diff b1f9bc4..56bc728 uniform_marked_polygon_bridge_v2.txt
diff --git a/uniform_marked_polygon_bridge_v2.txt b/uniform_marked_polygon_bridge_v2.txt
index 6ad8d64..a5bedf4 100644
--- a/uniform_marked_polygon_bridge_v2.txt
+++ b/uniform_marked_polygon_bridge_v2.txt
@@ -38,21 +38,21 @@ allowdisplaybreaks[2]
We prove quantitative estimates for the critical self-avoiding
walk on the honeycomb lattice, seen through three families of
geometric constraints. First, a bridge across a strip of
-height $h$ (a self-avoiding path from a fixed port on the bottom
+height h (a self-avoiding path from a fixed port on the bottom
cut to any port on the top cut) has total critical mass comparable
-to $h^{-1/4}$. The first moment of its length is at most
-$Ch^{13/12}$, and a positive fraction of the bridge mass is carried
-by paths of length at least $ch^{4/3}$. The bridge diameter tail is
-$C(1+x)^{-1/4}$.
-Second, on a cylinder of period $N$, the total critical mass
+to h^{-1/4}. The first moment of its length is at most
+Ch^{13/12}, and a positive fraction of the bridge mass is carried
+by paths of length at least ch^{4/3}. The bridge diameter tail is
+C(1+x)^{-1/4}.
+Second, on a cylinder of period N, the total critical mass
of a simple polygon passing through two specified honeycomb links
-at even axial displacement $d$ is at most $CN^{-4/3}(N/d)^{15/8}$,
-uniformly in the axis direction, in the total separation $d$ and
-in the ratio $N/d$.
+at even axial displacement d is at most CN^{-4/3}(N/d)^{15/8},
+uniformly in the axis direction, in the total separation d and
+in the ratio N/d.
Third, the positive partition function of systems of
mutually disjoint polygons enclosing a fixed triangular lattice
-vertex in a pure-sided hexagon of inradius $R$ is comparable to
-$R^{1/12}$.
+vertex in a pure-sided hexagon of inradius R is comparable to
+R^{1/12}.
The proofs come from a finite transfer-matrix calculation together
with a sequence of contour deformations of a
@@ -78,73 +78,73 @@ edge. The resulting graph -- vertices = triangle centres, edges =
(hexagonal) lattice. Every edge joins a black centre to a white
centre.
-A self-avoiding path $gamma$ is a finite sequence of
-distinct triangle centres $v_0,...,v_n$ with $v_j$ adjacent
-to $v_{j+1}$. Its length is $ell(gamma)=n+1$, the
+A self-avoiding path gamma is a finite sequence of
+distinct triangle centres v_0,...,v_n with v_j adjacent
+to v_{j+1}. Its length is ell(gamma)=n+1, the
number of visited centres. Its weight is
[
-w(gamma) = kappa^{ell(gamma)},qquad
+w(gamma) = kappa^{ell(gamma)},
kappa = (2cos(pi/8))^{-1} = 1/√(2+sqrt 2).
]
-The reciprocal $kappa^{-1}=√(2+sqrt2)$ is the
+The reciprocal kappa^{-1}=√(2+sqrt2) is the
connective constant of the honeycomb lattice; its
identification is a theorem of Duminil-Copin and Smirnov,
Ann. Math.~2012, proved by an exact local observable. Fix
-$kappa$. Summing $kappa^{ell(gamma)}$ over paths with prescribed
+kappa. Summing kappa^{ell(gamma)} over paths with prescribed
geometric constraints -- over all bridges of a strip, over
all polygons through two marks -- gives the three central sums.
The three theorems in plain words.:
Theorem thm:finite-bridges (bridges). Take a strip
-of height $h$ between two horizontal lattice cuts, and fix one
+of height h between two horizontal lattice cuts, and fix one
boundary port on the bottom cut. A bridge is a
self-avoiding path from that port to any port on the top cut,
-staying strictly between the cuts. Sum $kappa^{ell(gamma)}$
-over all bridges. Then the sum $B_h$ is comparable to
-$h^{-1/4}$ (positive lower and upper bounds), the first moment
-$sum ell(gamma)kappa^{ell(gamma)}$ is at most
-$Ch^{13/12}$, and a positive fraction of the mass lies on
-bridges of length at least $ch^{4/3}$. In words: crossing a
+staying strictly between the cuts. Sum kappa^{ell(gamma)}
+over all bridges. Then the sum B_h is comparable to
+h^{-1/4} (positive lower and upper bounds), the first moment
+sum ell(gamma)kappa^{ell(gamma)} is at most
+Ch^{13/12}, and a positive fraction of the mass lies on
+bridges of length at least ch^{4/3}. In words: crossing a
long strip is expensive but geometrical, so bridges are long (of
-order $h^{4/3}$) and rare (of order $h^{-1/4}$).
+order h^{4/3}) and rare (of order h^{-1/4}).
Theorem thm:two-marks (two marked edges). Put two
-specified honeycomb links on a cylinder of period $N$, with a
-particular even axial separation $d$ between them. The sum over
+specified honeycomb links on a cylinder of period N, with a
+particular even axial separation d between them. The sum over
simple polygons (closed self-avoiding curves on the cylinder) that
-pass through both marked links of $kappa^{ell(gamma)}$ is at
-most $CN^{-4/3}(N/d)^{15/8}$. In words: two marked edges cost an
-extra $N^{-4/3}$ beyond the baseline, and the further apart they
+pass through both marked links of kappa^{ell(gamma)} is at
+most CN^{-4/3}(N/d)^{15/8}. In words: two marked edges cost an
+extra N^{-4/3} beyond the baseline, and the further apart they
are (relatively), the more is gained. The bound is uniform in the
direction, so it is a genuine two-point correlation bound, not a
``homogeneous'' one.
Theorem thm:nesting (positive nesting). Around a
-triangular lattice vertex $y$ of a pure-sided hexagon of inradius
-$R$, sum over systems of mutually disjoint polygons in that hexagon
-that all enclose $y$, each with factor $2kappa^{ell}$. Then the
-resulting positive partition function $Z_2(y;R)$ grows like
-$R^{1/12}$, again with bounded constants. This ``positive''
+triangular lattice vertex y of a pure-sided hexagon of inradius
+R, sum over systems of mutually disjoint polygons in that hexagon
+that all enclose y, each with factor 2kappa^{ell}. Then the
+resulting positive partition function Z_2(y;R) grows like
+R^{1/12}, again with bounded constants. This ``positive''
comparison is the input needed to pass between positively weighted
path sums and positively weighted polygon sums.
-Notation.: Here $ell(gamma)$ is length in triangle
+Notation.: Here ell(gamma) is length in triangle
centres; for a simple polygon, length is equivalently the number of
edges. A boundary port of a tiled domain is a midpoint of a
boundary edge; paths enter and exit normally at ports. The auxiliary
-constants $L=pi/8$, $b=3L$, $C=2-sqrt2=4cos^2 b$ are eigenvalues
-of the row transfer matrix. For a triangular lattice vertex $y$ and
-$Rin d_0mathbb Z_bbZ_{>0}$ with $d_0=sqrt3/2$, $H(y;R)$ is the
-regular pure-sided hexagon of inradius $R$ centred at $y$.
+constants L=pi/8, b=3L, C=2-sqrt2=4cos^2 b are eigenvalues
+of the row transfer matrix. For a triangular lattice vertex y and
+Rin d_0mathbb Z_bbZ_{>0} with d_0=sqrt3/2, H(y;R) is the
+regular pure-sided hexagon of inradius R centred at y.
Introduction.:
The honeycomb lattice and its critical weight.:
-Every closed walk of $mathbb H$bbH$ drawn without self-intersection
+Every closed walk of mathbb HbbH drawn without self-intersection
encloses a union of triangles. A simple polygon
-$Lsubsetmathbb H$bbH$ is a simple closed curve of the honeycomb
+Lsubsetmathbb HbbH is a simple closed curve of the honeycomb
graph; it may be unbounded on a cylinder
(Section~sec:intro-cylinder), in which case it is
essential, or bounded, in which case it is
@@ -156,37 +156,37 @@ The critical weight of the self-avoiding walk is
kappa = (2cosfracpi8)^{-1} = 1/√(2+sqrt 2).
Duminil-Copin and Smirnov proved that
-$kappa^{-1}=√(2+sqrt2)$ is the connective constant of the
-honeycomb lattice. Fix $kappa$ throughout. A bridge contributes
-the $kappa$-mass over all self-avoiding paths fitting the
-constraint, a marked-polygon system contributes the $kappa$-mass
+kappa^{-1}=√(2+sqrt2) is the connective constant of the
+honeycomb lattice. Fix kappa throughout. A bridge contributes
+the kappa-mass over all self-avoiding paths fitting the
+constraint, a marked-polygon system contributes the kappa-mass
over all simple polygons through the marks, and a nesting system
-contributes the weighted $kappa$-mass over all mutually
+contributes the weighted kappa-mass over all mutually
disjoint sets of enclosing polygons.
Paths, bridges, polygons.:
-Let $D$ be a bounded tiled domain (a polygon in the triangular
+Let D be a bounded tiled domain (a polygon in the triangular
lattice with sides on the pure lattice cuts). A port of
-$D$ is the midpoint of a boundary edge; a self-avoiding path in
-$D$ begins and ends at ports, entering and exiting normally. Its
-weight is $kappa^{ell(gamma)}$, with $ell$ the number of
+D is the midpoint of a boundary edge; a self-avoiding path in
+D begins and ends at ports, entering and exiting normally. Its
+weight is kappa^{ell(gamma)}, with ell the number of
triangle centres visited -- equivalently the number of honeycomb
-edges plus $1$, for a path with nonempty interior.
+edges plus 1, for a path with nonempty interior.
A pure lattice cut is a line parallel to one of the three
edge directions of the triangular tiling. A pure-sided
domain is one whose sides lie on pure lattice cuts. For an
-integer $hge1$, let $mathcal B_h$calB_h$ be the set of bridges of
-$mathbb H$:bbH$: paths from a fixed bottom port to any port on the
-opposite horizontal cut at Euclidean distance $h d_0$ (where
-$d_0=sqrt3/2$ is the row spacing of the triangular lattice),
+integer hge1, let mathcal B_hcalB_h be the set of bridges of
+mathbb H:bbH: paths from a fixed bottom port to any port on the
+opposite horizontal cut at Euclidean distance h d_0 (where
+d_0=sqrt3/2 is the row spacing of the triangular lattice),
staying strictly between the cuts. Put
-B_h = sum_{gammainmathcal B_h}calB_h kappa^{ell(gamma)},qquad
+B_h = sum_{gammainmathcal B_h}calB_h kappa^{ell(gamma)},
B_h(E) = sum_{gammainmathcal B_h}calB_h mathbf 1_E(bf1_E(gamma)kappa^{ell(gamma)}.
-By lattice symmetry $B_h$ does not depend on the chosen bottom
+By lattice symmetry B_h does not depend on the chosen bottom
port or on the choice of pure lattice orientation. Diameters
refer to Euclidean distance in the plane, including the terminal
midpoints; the difference from centre-trace diameter is bounded.
@@ -194,69 +194,69 @@ midpoints; the difference from centre-trace diameter is bounded.
The three main theorems.:
[Finite bridge estimates]
-There exist constants $c,C>0$ such that for every sufficiently
-large integer $h$,
+There exist constants c,C>0 such that for every sufficiently
+large integer h,
c h^{-1/4} &le B_h,le, C h^{-1/4},
sum_{gammainmathcal B_h}calB_h ell(gamma)kappa^{ell(gamma)}
&le C h^{13/12},
B_h{ellge c h^{4/3}} &ge c h^{-1/4}.
-Moreover for every $hge1$ and $xge0$,
+Moreover for every hge1 and xge0,
B_h{diamgamma>x} le C(1+x)^{-1/4}.
Dividing (eq:bridge-first-moment) by (eq:bridge-mass)
-gives a conditional mean at most $Ch^{4/3}$. The positive mass
+gives a conditional mean at most Ch^{4/3}. The positive mass
in (eq:bridge-long-fraction) is uniform in height and
does not average over heights or prescribe the terminal port.
The second theorem uses two points, not one.
[Two axial marks]
-Let $e_1,e_2$ be unit lattice vectors at angle $60^circ$. Fix
-$P_1,P_2inmathbb Z_bbZ_{>0}$, $P=P_1e_1+P_2e_2$, $d=P_1+P_2$. Let
-$Esubset mathbb H$bbH$ be a honeycomb link crossing a tiling edge
-in either axial direction, and let $E+P$ be its translate by
-$P$. Suppose $d$ is even. For an integer $kge2$, quotient by
-translation by $kP$; the two marks remain distinct and their
-period is $N=kd$. Then
+Let e_1,e_2 be unit lattice vectors at angle 60^circ. Fix
+P_1,P_2inmathbb Z_bbZ_{>0}, P=P_1e_1+P_2e_2, d=P_1+P_2. Let
+Esubset mathbb HbbH be a honeycomb link crossing a tiling edge
+in either axial direction, and let E+P be its translate by
+P. Suppose d is even. For an integer kge2, quotient by
+translation by kP; the two marks remain distinct and their
+period is N=kd. Then
sum_{gammani E,,E+P} kappa^{ell(gamma)}
le C N^{-4/3}(N/d)^{15/8},
where the sum is over simple polygons in the quotient cylinder
-through both marked links, with $ell$ counting every visited
+through both marked links, with ell counting every visited
centre (so edges of the cylinder are counted once per lift).
-The constant $C$ is independent of $P_1,P_2,k$ and of the
+The constant C is independent of P_1,P_2,k and of the
choice of axial edge direction.
[Nesting and free boundary length]
-For a triangular lattice vertex $y$ and $Rin d_0mathbb Z_bbZ_{>0}$
-with $Rge1$, let $H(y;R)$ be the pure-sided regular hexagon of
-Euclidean inradius $R$ centred at $y$; the lattice restriction on
-$R$ and the centre is verified in
+For a triangular lattice vertex y and Rin d_0mathbb Z_bbZ_{>0}
+with Rge1, let H(y;R) be the pure-sided regular hexagon of
+Euclidean inradius R centred at y; the lattice restriction on
+R and the centre is verified in
Section~sec:hexagon-completion. For a finite tiled domain
-$D$ and a vertex $y$, put
+D and a vertex y, put
[
Z_D(y) = sum_{gamma_1,...,gamma_j disjoint in D}
2^j,kappa^{sum ell(gamma_i)},
]
-the sum over systems $(gamma_1,...,gamma_j)$ of mutually
-disjoint simple polygons, each enclosing $y$, and $jge0$ (so
-the empty system has weight 1). Then $Z_{H(y;R)}(y)$ is
-comparable to $R^{1/12}$ uniformly in $y$ and $R$. More
-generally, for any fixed $0<ale b<∞$ and any finite
-tiled domain $D$ with $B(y,aR)subset Dsubset B(y,bR)$,
-$Z_D(y)$ is comparable to $R^{1/12}$, uniformly over
-translations and pure-axis roundings of $D$. Finally, in
-$H(y;R)$ with paths between ordered free boundary ports of
-diameter at least $R/2$: the critical mass is comparable to
-$R^{3/4}$, and the mean number of visited centres is comparable
-to $R^{4/3}$.
+the sum over systems (gamma_1,...,gamma_j) of mutually
+disjoint simple polygons, each enclosing y, and jge0 (so
+the empty system has weight 1). Then Z_{H(y;R)}(y) is
+comparable to R^{1/12} uniformly in y and R. More
+generally, for any fixed 0<ale b<∞ and any finite
+tiled domain D with B(y,aR)subset Dsubset B(y,bR),
+Z_D(y) is comparable to R^{1/12}, uniformly over
+translations and pure-axis roundings of D. Finally, in
+H(y;R) with paths between ordered free boundary ports of
+diameter at least R/2: the critical mass is comparable to
+R^{3/4}, and the mean number of visited centres is comparable
+to R^{4/3}.
Context and history.:
@@ -264,27 +264,27 @@ Context and history.:
The honeycomb self-avoiding walk has a long history, and three
results sit directly behind Theorem~thm:finite-bridges.
Nienhuis~[cite] predicted the critical point and
-critical exponents from the dilute $O(n)$ model. Duminil-Copin
+critical exponents from the dilute O(n) model. Duminil-Copin
and Smirnov~[cite] proved the exact
connective constant. Beaton, Bousquet-M'elou, de Gier,
Duminil-Copin and Guttmann~[cite] proved the
-strip-crossing mass goes to zero as $hto∞$. Glazman and
+strip-crossing mass goes to zero as hto∞. Glazman and
Manolescu~[cite] obtained a logarithmic
bound along a subsequence and identified the predicted
-$h^{-1/4}$ decay; Krachun and Panagiotis~[cite]
-proved a polynomial upper bound with exponent $10^{-10}$.
+h^{-1/4} decay; Krachun and Panagiotis~[cite]
+proved a polynomial upper bound with exponent 10^{-10}.
Theorem~thm:finite-bridges gives the predicted decay with
fixed constants, together with first-moment and positive-mass
length estimates on the same finite bridge measure.
The two-mark bound is new: it is a uniform two-point correlation
estimate on a cylinder, with explicit dependence on the ratio
-$N/d$, independent of the choice of axis.
+N/d, independent of the choice of axis.
`A~propos the analytic structure, the transfer matrix used in
Section~c6:strip-transfer is a specialization of the
integrable dilute-loop algebra of Grimm and
Pearce~[cite], and the polynomial-exchange
-formalism extends the dense-$O(1)$ construction of Di Francesco
+formalism extends the dense-O(1) construction of Di Francesco
and Zinn-Justin~[cite] and the dilute
open-boundary work of Garbali and
Nienhuis~[cite]
@@ -296,12 +296,12 @@ Segal~[cite] supplies the
current-expansion framework; the thermal trace estimates and
contour deformations that enter here are done explicitly in
Section~c6:gas. Precedents for the interior-multiplicity
-argument and the $O(n=1)$ bulk behaviour include the work of
+argument and the O(n=1) bulk behaviour include the work of
Zhou and Batchelor~[cite].
The geometry side has an apparent ``obstruction'': joining a
chord to an exterior connector would na"ively lose a factor of
-$kappa$ at each join, and a naive sum over joins would
+kappa at each join, and a naive sum over joins would
diverge. The solution adopted in Section~sec:sewing is to
vary the intermediate cuts and use a second-moment
estimate; the difference between two cut choices produces
@@ -348,47 +348,47 @@ A finite formula for strip crossing mass.:
% ============================================================
-Goal of this section. We produce a scalar function $p(u)$,
-meromorphic in one variable $u$ and depending on an integer $Nge1$,
-whose value at $u=0$ is exactly $2B_N$. The function $p$ has enough
+Goal of this section. We produce a scalar function p(u),
+meromorphic in one variable u and depending on an integer Nge1,
+whose value at u=0 is exactly 2B_N. The function p has enough
analytic structure that it is determined by finitely many linear
conditions, which Section~c6:strip-asymptotics will estimate
directly.
-The underlying object is a commuting family $T(t)$ of matrices on
-$N$-site spin space, called the double-row transfer matrix. At
-a specific value $t=gamma$ of the spectral parameter, $T(gamma)$
-has a rank-one projection as $mto∞$ limit for its $m$-th power;
-this limit is the physical bridge measure of the height-$2m$
+The underlying object is a commuting family T(t) of matrices on
+N-site spin space, called the double-row transfer matrix. At
+a specific value t=gamma of the spectral parameter, T(gamma)
+has a rank-one projection as mto∞ limit for its m-th power;
+this limit is the physical bridge measure of the height-2m
rectangle. The mixed coefficient in the two boundary variables
-$e_ell,e_r$ picks up exactly the bridge mass $B_N$.
+e_ell,e_r picks up exactly the bridge mass B_N.
Constants and setup.:
Throughout fix
[
-L=(pi)/(8),quad S=sin(2L)=frac 1{sqrt2},quad b=3L,quad
-gamma=frac b2,quad kappa=(2cos L)^{-1}=1/√(2+sqrt2).
+L=(pi)/(8), S=sin(2L)=frac 1{sqrt2}, b=3L,
+gamma=frac b2, kappa=(2cos L)^{-1}=1/√(2+sqrt2).
]
-We write $t_j=sin(u+jL)$ when the argument $u$ is clear from context.
+We write t_j=sin(u+jL) when the argument u is clear from context.
Two elementary sine identities are used repeatedly:
-t_{j-1}+t_{j+1} = 2cos L, t_j = t_j/kappa,qquad
+t_{j-1}+t_{j+1} = 2cos L, t_j = t_j/kappa,
t_{j-2}t_{j+2} = t_j^2 - S^2,
-both direct consequences of $sin(a+b)+sin(a-b)=2sin acos b$ and
-$sin(a+b)sin(a-b)=sin^2 a-sin^2 b$ together with $sin(2L)=1/sqrt2$.
-Also $sin(t+pi)=-sin t$, so any expression $sin(u+jL)$ is
-periodic or antiperiodic mod $pi$. At $u=2L$: $sin(2L+2L)=cos(0)=1$,
-and $sin(4L-2L)=sin(2L)=1/sqrt2$.
+both direct consequences of sin(a+b)+sin(a-b)=2sin acos b and
+sin(a+b)sin(a-b)=sin^2 a-sin^2 b together with sin(2L)=1/sqrt2.
+Also sin(t+pi)=-sin t, so any expression sin(u+jL) is
+periodic or antiperiodic mod pi. At u=2L: sin(2L+2L)=cos(0)=1,
+and sin(4L-2L)=sin(2L)=1/sqrt2.
-The local $R$-matrix.:
+The local R-matrix.:
-Let $V=mathbb C^3$bbC^3$ have basis ${|+>,|0>,|->}$,
-interpreted as spins $+1,0,-1$. Set $s = e^{2iL} = e^{ipi/4}$ so
-$s^4=-1$ and $s^2+s^{-2}=0$. Put $Q(u)=sin(2L+u)sin(b+u)$ and
+Let V=mathbb C^3bbC^3 have basis {|+>,|0>,|->},
+interpreted as spins +1,0,-1. Set s = e^{2iL} = e^{ipi/4} so
+s^4=-1 and s^2+s^{-2}=0. Put Q(u)=sin(2L+u)sin(b+u) and
define
@@ -399,7 +399,7 @@ c(u)&=(sin(b-u)sin(2L-u))/(Q(u)),&
d(u)&=(sin u sin(u-L))/(Q(u)).
-Then $R(u)inEnd(V ⊗ V)$ is defined by
+Then R(u)inEnd(V ⊗ V) is defined by
&R(++,++)=R(--,--)=c(u),
@@ -417,32 +417,32 @@ c(u)+d(u)s^2&s,h(u)&d(u)
.
-All other entries vanish: $R(pmpm,mpmp)$, $R(0pm,pm-)$, etc. are
+All other entries vanish: R(pmpm,mpmp), R(0pm,pm-), etc. are
zero. Summation over intermediate spins is implied.
-Structural identities of $R$.:
+Structural identities of R.:
-Let $P$ denote the swap $P(a ⊗ b)=b ⊗ a$,
-$J=diag(-1,1,-1)$, $F=(0&0&10&1&01&0&0)$
+Let P denote the swap P(a ⊗ b)=b ⊗ a,
+J=diag(-1,1,-1), F=(0&0&10&1&01&0&0)
(spin flip), and
Ω=sum_{pin{-,0,+}} s^{-p},|p,-p>in V ⊗ V.
-Define moreover the rank-two operator $WinHom(V,V ⊗ V)$ by
+Define moreover the rank-two operator WinHom(V,V ⊗ V) by
-W(|+>)=kappa(|+,0>+|0,+>),quad
-W(|->)=kappa(|-,0>+|0,->),quad
+W(|+>)=kappa(|+,0>+|0,+>),
+W(|->)=kappa(|-,0>+|0,->),
W(|0>)=|00>+kappasum_{pin{-,+}} s^{-p}|p,-p>,
-and $Z=W^*=barbar W^TW^TinHom(V ⊗ V,V)$.
+and Z=W^*=barbar W^TW^TinHom(V ⊗ V,V).
-paragraph*{The identities.} We use the following, verified by direct
+The identities. We use the following, verified by direct
substitution in (eq:R-coeffs)--(eq:R-matrix).
-[$R$-matrix identities]
+[R-matrix identities]
-For all complex $u$ away from poles,
+For all complex u away from poles,
R_{21}(u) &= R_{12}^T(u),&(transpose)
R_{12}(u) R_{12}^T(-u) &= I,&(unitarity)
@@ -455,43 +455,43 @@ R_{12}(2L) &= W Z.&(rank-two at 2L)
Proof.
-Transpose and unitarity. $R(u)_{ij,kl}=R(u)_{kl,ij}$ by
-inspection (symmetric entries). Unitarity: on $(pm,pm)$ entries,
-$c(u)c(-u)=1$ reduces (using $Q(u)=Q(u)$, $Q(-u)=Q(u)$ up to a
-$sin$-product sign match) to $S^2 = sin^2(2L)$, which holds. On
-$(+,0)$-type entries, $v(u)v(-u)+a(u)a(-u)=1$ reduces to the same
-identity. On the $(+-,00,-+)$ block, the six nonzero entries each give
-sine-product identities; for instance $d(u)d(-u)+c(u)c(-u)+c(u)d(u)s^2
-+ c(-u)d(-u)s^{-2}$ (a linear combination appearing at
-$(+-, -+)$) is $d(u)d(-u)+c(u)c(-u)$ by $s^2+s^{-2}=0$, which then
-reduces to the $(pm,pm)$ check.
-
-Pi-periodicity. $Q(u+pi)=-Q(u)$; $a(u+pi)=-a(u)$,
-$h(u+pi)=-h(u)$ (numerator flips sign), $c(u+pi)=c(u)$,
-$d(u+pi)=d(u)$, $v(u+pi)=v(u)$. Thus on $(pm,pm)$, $R$ is
-pi-periodic; on $(+,0)$-type, $R$ is antisymmetric under $umapsto
-u+pi$, matching the $J$ insertion on either side; on $(+-,00,-+)$,
-the off-diagonal terms $h/s$ flip sign, and the $J_1$ on either side
+Transpose and unitarity. R(u)_{ij,kl}=R(u)_{kl,ij} by
+inspection (symmetric entries). Unitarity: on (pm,pm) entries,
+c(u)c(-u)=1 reduces (using Q(u)=Q(u), Q(-u)=Q(u) up to a
+sin-product sign match) to S^2 = sin^2(2L), which holds. On
+(+,0)-type entries, v(u)v(-u)+a(u)a(-u)=1 reduces to the same
+identity. On the (+-,00,-+) block, the six nonzero entries each give
+sine-product identities; for instance d(u)d(-u)+c(u)c(-u)+c(u)d(u)s^2
++ c(-u)d(-u)s^{-2} (a linear combination appearing at
+(+-, -+)) is d(u)d(-u)+c(u)c(-u) by s^2+s^{-2}=0, which then
+reduces to the (pm,pm) check.
+
+Pi-periodicity. Q(u+pi)=-Q(u); a(u+pi)=-a(u),
+h(u+pi)=-h(u) (numerator flips sign), c(u+pi)=c(u),
+d(u+pi)=d(u), v(u+pi)=v(u). Thus on (pm,pm), R is
+pi-periodic; on (+,0)-type, R is antisymmetric under umapsto
+u+pi, matching the J insertion on either side; on (+-,00,-+),
+the off-diagonal terms h/s flip sign, and the J_1 on either side
supplies the same flip. This identifies both sides.
-Crossing. Since $Q(b-u)=Q(u)$, we have
+Crossing. Since Q(b-u)=Q(u), we have
-a(b-u)&=Ssin(u)big/Q(u)=a(u),quad
+a(b-u)&=Ssin(u)big/Q(u)=a(u),
h(b-u)=Ssin(b-u)big/Q(u)=h(u),
v(b-u)&=sin(u)sin(b-u)/Q(u)=v(u),
c(b-u)&=sin u sin(u-2L)/Q(u) = c(u)+textstyle...
-The last is $c(b-u) = sin u sin(u-2L)/Q(u)$. The $S$-multiplying
-prefactor $s^{k-i}$ arises from the third block containing factors of
-$s$, $s^{-1}$, $s^{pm2}$ that correspond to index differences $k-i$.
+The last is c(b-u) = sin u sin(u-2L)/Q(u). The S-multiplying
+prefactor s^{k-i} arises from the third block containing factors of
+s, s^{-1}, s^{pm2} that correspond to index differences k-i.
Each entry of (eq:R-crossing) follows.
-Special values at $0$, $b$, $2L$. At $u=0$: $a=h=d=0$,
-$v=c=1$; $R$ reduces to permutation $P$. At $u=b$: $Q(b)=sin(4L)sin(2b)
-= 1 cdot sin(6L) = sin(pi/4+pi/2) = sin(3pi/4) = S$;
-direct check gives $R(b)=ΩbarbarΩ^T$.Ω^T$. At $u=2L$:
-$Q(2L)=sin(4L)sin(2L+3L)=sin(4L)sin(5L)=1cdot S$; direct check
-gives $R(2L)=WZ$.
+Special values at 0, b, 2L. At u=0: a=h=d=0,
+v=c=1; R reduces to permutation P. At u=b: Q(b)=sin(4L)sin(2b)
+= 1 cdot sin(6L) = sin(pi/4+pi/2) = sin(3pi/4) = S;
+direct check gives R(b)=ΩbarbarΩ^T.Ω^T. At u=2L:
+Q(2L)=sin(4L)sin(2L+3L)=sin(4L)sin(5L)=1cdot S; direct check
+gives R(2L)=WZ.
The fusion equation.:
@@ -500,12 +500,12 @@ The fusion equation.:
The key insertion identity:
[Basic fusion]
-Let $f$ index a fused spin. Then on $V_1 ⊗ V_2 ⊗ V_3$,
+Let f index a fused spin. Then on V_1 ⊗ V_2 ⊗ V_3,
R_{23}(u-L),R_{13}(u+L),W_{12} = W_{12},R_{f3}(u).
-Here $W_{12}$ acts on slots $1,2$ with fused output $f$, and $R_{f3}$
-acts on $f ⊗ V_3$.
+Here W_{12} acts on slots 1,2 with fused output f, and R_{f3}
+acts on f ⊗ V_3.
Proof.
@@ -514,62 +514,62 @@ Transposing gives the adjoint form
Z_{12},R_{13}(u+L),R_{23}(u-L) = R_{f3}(u),Z_{12}.
We prove (eq:R-fusion). Multiply both sides on the right by
-$R_{32}(L-u)=R_{23}(L-u)^{-1}$ (unitarily, using $R_{32}=R_{23}^T$ and
+R_{32}(L-u)=R_{23}(L-u)^{-1} (unitarily, using R_{32}=R_{23}^T and
(eq:R-unitarity)). The required identity becomes
[
R_{23}(u-L),R_{13}(u+L),W_{12},R_{32}(L-u) = W_{12}.
]
-Both sides are analytic in $u$ away from the poles of $R$; it suffices
+Both sides are analytic in u away from the poles of R; it suffices
to check on a spanning set of input spin configurations on the three
-spaces $(f,3)$ with the "output" of $W_{12}$ being a pair of spins.
+spaces (f,3) with the "output" of W_{12} being a pair of spins.
-The pairs of spins we need to consider are (up to spin flip): $(+,+)$
-and $(+,-)$ and $(+,0)$ and $(0,0)$ and $(+,f)$-type with $fin{+,0,-}$.
-Each case reduces to sine-product identities in $t_j=sin(u+jL)$. The
-specific list, writing primes for argument $L-u$, $Q'=Q(L-u)$,
-$Q_+=Q(u+L)$, and using $QQ'/Q_+=-t_{-3}t_2$ (a direct computation from
-$sin(cdot)$ products), is:
+The pairs of spins we need to consider are (up to spin flip): (+,+)
+and (+,-) and (+,0) and (0,0) and (+,f)-type with fin{+,0,-}.
+Each case reduces to sine-product identities in t_j=sin(u+jL). The
+specific list, writing primes for argument L-u, Q'=Q(L-u),
+Q_+=Q(u+L), and using QQ'/Q_+=-t_{-3}t_2 (a direct computation from
+sin(cdot) products), is:
-On $(+,+)$-input:
+On (+,+)-input:
-cv' = c_+,quad cc'=v_+,quad ca'=a_+.
+cv' = c_+, cc'=v_+, ca'=a_+.
-On $(+,0)$-input and $(0,+)$-input (spin flips are analogous):
+On (+,0)-input and (0,+)-input (spin flips are analogous):
-v+ah'=v_+,quad vv'+aa'/kappa=1,quad va'+av'/kappa=a_+,quad
-vh'+ad'=0,quad ac'=h_+;
-kappa(a+vh')=a_+,quad av'+va'/kappa=h_+,quad
-kappa aa'+vv'=v_+,quad ah'+vd'=c_+,quad vc'=d_+.
+v+ah'=v_+, vv'+aa'/kappa=1, va'+av'/kappa=a_+,
+vh'+ad'=0, ac'=h_+;
+kappa(a+vh')=a_+, av'+va'/kappa=h_+,
+kappa aa'+vv'=v_+, ah'+vd'=c_+, vc'=d_+.
-On $(00)$-input:
+On (00)-input:
-kappa(a'+v'h)=h_+,quad v'+a'h=v_+,quad h'+kappa h c'=kappa a_+.
+kappa(a'+v'h)=h_+, v'+a'h=v_+, h'+kappa h c'=kappa a_+.
-On $(+-)$-input:
+On (+-)-input:
-h+kappa c h'=kappa h_+,quad hh'+kappa(cd'+c'd)=0,quad
-dv'+ha'=d_+,quad da'+hv'=0,
+h+kappa c h'=kappa h_+, hh'+kappa(cd'+c'd)=0,
+dv'+ha'=d_+, da'+hv'=0,
-together with the $(+,+)$-input relations applied to the $(-,-)$ pair.
+together with the (+,+)-input relations applied to the (-,-) pair.
Each of the above identities reduces to combinations of the six
elementary sine products
-t_{-2}t_2 = t_0^2 - S^2,quad
-t_{j-1}+t_{j+1}=t_j/kappa,quad
+t_{-2}t_2 = t_0^2 - S^2,
+t_{j-1}+t_{j+1}=t_j/kappa,
t_{-3}t_4=t_0t_1 - S^2/kappa,
-t_3t_4 = S^2/kappa - t_0 t_{-1},quad
-t_0t_{-1}-t_{-2}t_1=kappa S^2,quad
+t_3t_4 = S^2/kappa - t_0 t_{-1},
+t_0t_{-1}-t_{-2}t_1=kappa S^2,
t_2t_1 - t_{-2}t_{-1}=t_0t_4/kappa.
-For example, "$v+ah'=v_+$" is exactly $-t_{-3}t_0 + S t_{-3}S
-t_{-1}/(Q'Q) stackrel{?}{=} -t_1t_{-2}/Q_+$; cross-multiplying by
-$QQ_+$ reduces (using $QQ'/Q_+=-t_{-3}t_2$) to a case of
+For example, "v+ah'=v_+" is exactly -t_{-3}t_0 + S t_{-3}S
+t_{-1}/(Q'Q) stackrel{?}{=} -t_1t_{-2}/Q_+; cross-multiplying by
+QQ_+ reduces (using QQ'/Q_+=-t_{-3}t_2) to a case of
(eq:fusion-aux). All remaining cases are identical.
@@ -577,54 +577,54 @@ The Yang--Baxter equation.:
[Yang--Baxter]
-For all $u,v$,
+For all u,v,
R_{12}(u-v),R_{13}(u),R_{23}(v) = R_{23}(v),R_{13}(u),R_{12}(u-v).
Proof.
-Fix $v=0$ (general $v$ follows by shift). At $u-v=0$, both sides are
-$P R R$ vs $R R P$, which coincide because $R$ is $P$-symmetric on the
+Fix v=0 (general v follows by shift). At u-v=0, both sides are
+P R R vs R R P, which coincide because R is P-symmetric on the
symmetric block and antisymmetric on the antisymmetric block (checked
-from (eq:R-matrix)). At $u-v=2L$, use $R(2L)=WZ$ and
-(eq:R-fusion)+(eq:R-fusion-adjoint): the $W$ on the left
-absorbs one $R$ on each side, leaving $Id$ on the fused
-space, and the $Z$ similarly closes to identity on the right. At
-$u-v=b$, use $R(b)=ΩbarbarΩ^T$:Ω^T$: $W$ now inserts $Ω$, and
-both sides collapse to $Ω$ paired with itself (with the adjoint
+from (eq:R-matrix)). At u-v=2L, use R(2L)=WZ and
+(eq:R-fusion)+(eq:R-fusion-adjoint): the W on the left
+absorbs one R on each side, leaving Id on the fused
+space, and the Z similarly closes to identity on the right. At
+u-v=b, use R(b)=ΩbarbarΩ^T:Ω^T: W now inserts Ω, and
+both sides collapse to Ω paired with itself (with the adjoint
fusion closing the loop).
-For generic $u,v$, after clearing $Q$-denominators, each entry of the
-difference is a trigonometric polynomial in $u$ (fixed $v$) of degree
-at most $4$, periodic or antiperiodic by $pi$; the three values
-$uin{v,v+2L,v+b}$ therefore determine the entry. This proves
+For generic u,v, after clearing Q-denominators, each entry of the
+difference is a trigonometric polynomial in u (fixed v) of degree
+at most 4, periodic or antiperiodic by pi; the three values
+uin{v,v+2L,v+b} therefore determine the entry. This proves
(eq:YBE) identically.
-The boundary matrix $K_e(t)$.:
+The boundary matrix K_e(t).:
The row transfer also needs a boundary matrix that closes each
-row into a double row. For a formal variable $e$ with $e^2=0$, set
+row into a double row. For a formal variable e with e^2=0, set
-k(t)=(sin(gamma-t))/(sin(gamma+t)),qquad
+k(t)=(sin(gamma-t))/(sin(gamma+t)),
l(t)=(sin(2t))/(sin(gamma+t)),
-$D_e(t)=diag(k(t),1,k(t))+ e,l(t)(E_{+,0}+s E_{0,-})in
-End(V)$, and
+D_e(t)=diag(k(t),1,k(t))+ e,l(t)(E_{+,0}+s E_{0,-})in
+End(V), and
-K_e(t)inHom(V,V ⊗ V),qquad K_e(t)_{,p,-m}
+K_e(t)inHom(V,V ⊗ V), K_e(t)_{,p,-m}
=s^m(D_e(t))_{pm}.
-That is, $K_e(t)$ makes the $+/-$ boundary identifications via the $s$
-weight while imposing $D_e$ on the surviving pair.
+That is, K_e(t) makes the +/- boundary identifications via the s
+weight while imposing D_e on the surviving pair.
[Boundary fusion]
-For all complex $t$, with $r=b-t$:
+For all complex t, with r=b-t:
D_e(t)D_e(-t) &= I,
R(t-r) K_e(t) &= P K_e(r),
@@ -634,136 +634,135 @@ W^T D_{e,2}(t+L),R(2t),D_{e,1}(t-L) &= D_e(t),barbar W^T.W^T.
Proof.
-Unitarity. $k(-t)=sin(gamma+t)/sin(gamma-t)=1/k(t)$ (oddness of $sin$). Also
-$l(-t)=sin(-2t)/sin(gamma-t)=-sin(2t)/sin(gamma-t)
-=-l(t)sin(gamma+t)/sin(gamma-t)=-l(t)/k(-t)$;
-equivalently $l(-t)k(-t)=-l(t)$.
-Now the diagonal of $D_e(t)D_e(-t)$ is
-$diag(k(t)k(-t),1,k(t)k(-t))=I$. Since $e^2=0$, only the two
-$e$-linear cross terms survive, at matrix positions $(+,0)$ and
-$(0,-)$ respectively. Coefficient of $(+,0)$: $k(t)l(-t)+l(t)$.
-Coefficient of $(0,-)$: $s,[k(t)l(-t)+l(t)]$. Both vanish by
-$l(-t)=-l(t)/k(t)$. Hence $D_e(t)D_e(-t)=I$.
-
-Swapping K. On total spin $0$ (component $barbarΩ$),Ω$), the
-check reduces to $c(t-r) k(t)+h(t-r)=k(r)$; on total spin $1$ it
-reduces to $(v+a)(t-r)l(t)=l(r)$. Writing $y=t-gamma$, these follow
+Unitarity. k(-t)=sin(gamma+t)/sin(gamma-t)=1/k(t) (oddness of sin). Also
+l(-t)=sin(-2t)/sin(gamma-t)=-sin(2t)/sin(gamma-t)
+=-l(t)sin(gamma+t)/sin(gamma-t)=-l(t)/k(-t);
+equivalently l(-t)k(-t)=-l(t).
+Now the diagonal of D_e(t)D_e(-t) is
+diag(k(t)k(-t),1,k(t)k(-t))=I. Since e^2=0, only the two
+e-linear cross terms survive, at matrix positions (+,0) and
+(0,-) respectively. Coefficient of (+,0): k(t)l(-t)+l(t).
+Coefficient of (0,-): s,[k(t)l(-t)+l(t)]. Both vanish by
+l(-t)=-l(t)/k(t). Hence D_e(t)D_e(-t)=I.
+
+Swapping K. On total spin 0 (component barbarΩ),Ω), the
+check reduces to c(t-r) k(t)+h(t-r)=k(r); on total spin 1 it
+reduces to (v+a)(t-r)l(t)=l(r). Writing y=t-gamma, these follow
from the two sine identities
-sin(L-y)sin(b-2y)+sin(L+y)sin(b+2y)=Scos y,qquad
+sin(L-y)sin(b-2y)+sin(L+y)sin(b+2y)=Scos y,
cos(Lpm y)=sin(bmp y).
Fusion below. We sketch. Transposing the second fusion gives the
-fourth (using $Fbarbar D_eD_e F = D_{e/s}^T$ at formal real $e$). For the
-third, (eq:K-fusion-below), evaluate at $u=2t$ and use
-$Q(2t)=sin(4L)sin(2t+3L)cdot...$; on the three columns of
-$D_{e,2}R(2t)D_{e,1}barbar W$W$ the constant term in $e$ reduces to
-$v k_+ + a = k$ and the linear term to $k_-(a k_+ + v)=k$ and
-$k_-(h+kappa c k_+)=kappa$. Writing
-$H_pm=sin(gammapm t)$, $G_pm=cos(gammapm t)$,
-$F_pm=sin(L/2pm t)$, $P_pm=sin(2Lpm 2t)$, $T_0=sin(2t)$,
-substituting $k_+=F_-/G_-$, $k_-=G_+/F_+$, $l_+=P_+/G_-$,
-$l_-=-P_-/F_+$ gives the sine identities
-
-
-P_+ G_+-S G_-=T_0 F_-,quad
-P_+F_+-S F_-=T_0 G_-,quad
+fourth (using Fbarbar D_eD_e F = D_{e/s}^T at formal real e). For the
+third, (eq:K-fusion-below), evaluate at u=2t and use
+Q(2t)=sin(4L)sin(2t+3L)cdot...; on the three columns of
+D_{e,2}R(2t)D_{e,1}barbar WW the constant term in e reduces to
+v k_+ + a = k and the linear term to k_-(a k_+ + v)=k and
+k_-(h+kappa c k_+)=kappa. Writing
+H_pm=sin(gammapm t), G_pm=cos(gammapm t),
+F_pm=sin(L/2pm t), P_pm=sin(2Lpm 2t), T_0=sin(2t),
+substituting k_+=F_-/G_-, k_-=G_+/F_+, l_+=P_+/G_-,
+l_-=-P_-/F_+ gives the sine identities
+
+
+P_+ G_+-S G_-=T_0 F_-,
+P_+F_+-S F_-=T_0 G_-,
2(H_+ F_+ P_+ -H_-F_-P_-) = S T_0/kappa,
-P_- H_+-S H_-=T_0 F_-,quad kappa H_- -F_-=kappa F_+,quad
+P_- H_+-S H_-=T_0 F_-, kappa H_- -F_-=kappa F_+,
H_-=kappa(F_-+G_+),
-S+sin(2t-L)=2F_+G_-,quad
+S+sin(2t-L)=2F_+G_-,
P_+ H_+ G_+-P_- H_- G_-=cos L,T_0cos(2t).
-Each is a direct substitution using $8L=pi$ and $sin(2L)=1/sqrt2$.
+Each is a direct substitution using 8L=pi and sin(2L)=1/sqrt2.
-The double-row operator $T(t)$.:
+The double-row operator T(t).:
-For an integer $Nge1$, let $V_{sites}=V^{ ⊗ N}$ with
+For an integer Nge1, let V_{sites}=V^{ ⊗ N} with
U_alpha(t) = big(D_e(t),R(t+zeta_{a_1})... R(t+zeta_{a_N})big)_{a_1... a_N;b_1... b_N}
-where $alphain{1,2}$ labels the two rows and $zeta_j$ are the
-$N$ fixed rapidities. Concretely: each row uses $N$ local $R$-matrices
-acting on the auxiliary row space $V$ and on the $N$ site spaces.
+where alphain{1,2} labels the two rows and zeta_j are the
+N fixed rapidities. Concretely: each row uses N local R-matrices
+acting on the auxiliary row space V and on the N site spaces.
-The full operator on $V ⊗ V ⊗ V^{ ⊗ N} ⊗ V ⊗ V$
-(the two boundary auxiliary spaces plus the $N$ site spaces) is
+The full operator on V ⊗ V ⊗ V^{ ⊗ N} ⊗ V ⊗ V
+(the two boundary auxiliary spaces plus the N site spaces) is
-T(t) = K_{e_ell}(r)_{AB}^T,U_A(t),U_B(r),K_{e_r}(t)_{AB},qquad r=b-t,
+T(t) = K_{e_ell}(r)_{AB}^T,U_A(t),U_B(r),K_{e_r}(t)_{AB}, r=b-t,
-where $e_ell$, $e_r$ are formal nilpotent parameters ($e_ell^2=e_r^2=0$).
-The transposed $K$ acts as a covector on the left.
+where e_ell, e_r are formal nilpotent parameters (e_ell^2=e_r^2=0).
+The transposed K acts as a covector on the left.
-$T(t)$ has the following properties, which are what we will use.
+T(t) has the following properties, which are what we will use.
-[Structure of $T(t)$]
-
-item $T(t)$ is $pi$-periodic: $T(t+pi)=T(t)$.
-item $T(t)$ is symmetric under interchange of the two rows: $T(t)=T(r)$.
-item $T(0)=T(4L)=I$ on the physical (total-spin-zero) sector.
+[Structure of T(t)]
+ T(t) is pi-periodic: T(t+pi)=T(t).
+ T(t) is symmetric under interchange of the two rows: T(t)=T(r).
+ T(0)=T(4L)=I on the physical (total-spin-zero) sector.
Proof.
-(1) $K_e(t+pi)=J_1 J_2 K_e(t)$, using $k(t+pi)=k(t)$, $l(t+pi)=l(t)$
-(period $pi$ of sines with $gamma=3L$). Then $J_1 J_2 R(t+pi+zeta)
-= R(t+zeta)J_1 J_2$ by (eq:R-pi-periodic), and $J_1 J_2$ cancels
+(1) K_e(t+pi)=J_1 J_2 K_e(t), using k(t+pi)=k(t), l(t+pi)=l(t)
+(period pi of sines with gamma=3L). Then J_1 J_2 R(t+pi+zeta)
+= R(t+zeta)J_1 J_2 by (eq:R-pi-periodic), and J_1 J_2 cancels
with its transpose.
(2) Row interchange: Yang--Baxter gives
-$R_{ij}(t-r)U_i(t)U_j(r) = U_j(r)U_i(u) R_{ij}(t-r)$ up to labels; for
-$T$ this reduces to the input and output swap identities
-$R(t-r)K_e(t) = P K_e(r)$ and the $r,t$ version. So $T$ is symmetric.
-
-(3) $T(0)=I$: at $t=0$ we have $r=b$ and $k(0)=1$, $l(0)=0$, so
-$K_e(0)=barbarΩ$.Ω$. Both local $R$'s, $R(0)=P$ and $R(b)=ΩbarbarΩ^T$,Ω^T$,
-act as rank-one contractions on the input $barbarΩ$,Ω$, giving
-$(T(0))_{∅,∅}=1$ (in fact $T(0)=I$). For
-$T(4L)$: use $T(t+pi)=T(t)$ from (1) to rewrite the second row's
-parameter $r=-L$ as $r+pi=7L$; then $t-r=4L-7L=-b$, and the
-swap structure of the two $R$'s at parameters differing by $b$
+R_{ij}(t-r)U_i(t)U_j(r) = U_j(r)U_i(u) R_{ij}(t-r) up to labels; for
+T this reduces to the input and output swap identities
+R(t-r)K_e(t) = P K_e(r) and the r,t version. So T is symmetric.
+
+(3) T(0)=I: at t=0 we have r=b and k(0)=1, l(0)=0, so
+K_e(0)=barbarΩ.Ω. Both local R's, R(0)=P and R(b)=ΩbarbarΩ^T,Ω^T,
+act as rank-one contractions on the input barbarΩ,Ω, giving
+(T(0))_{∅,∅}=1 (in fact T(0)=I). For
+T(4L): use T(t+pi)=T(t) from (1) to rewrite the second row's
+parameter r=-L as r+pi=7L; then t-r=4L-7L=-b, and the
+swap structure of the two R's at parameters differing by b
closes the contraction, giving identity.
-Commutation of $T(t)$ and $T(y)$.:
+Commutation of T(t) and T(y).:
[Commuting double rows]
-For all $t,y$ in $mathbb C$bbC$ away from poles,
+For all t,y in mathbb CbbC away from poles,
T(t)T(y) = T(y)T(t).
Proof.
-Take labels $A,B,C,D'$ of parameters $t,r,y,b-y$ respectively.
-To interchange the pairs $BC$ and $AC$ (resp. $BD'$ and $AD'$), apply
-$R_{AC}(t-y)$ and $R_{BD'}^T(t-y)$ and their $b-y$ analogues. Invariance
+Take labels A,B,C,D' of parameters t,r,y,b-y respectively.
+To interchange the pairs BC and AC (resp. BD' and AD'), apply
+R_{AC}(t-y) and R_{BD'}^T(t-y) and their b-y analogues. Invariance
of the right boundary vector
-$V=K_{e_r}(t)_{AB}K_{e_r}(y)_{CD'}$ reduces (using
+V=K_{e_r}(t)_{AB}K_{e_r}(y)_{CD'} reduces (using
(eq:R-crossing)) to the reflection identity
R(t-y)D_{e,1}(t)R(t+y)^T D_{e,2}(y)
=D_{e,2}(y)R(t+y)D_{e,1}(t)R(t-y)^T.
-The left boundary works identically by transposing. So $T(t)T(y)=T(y)T(t)$.
+The left boundary works identically by transposing. So T(t)T(y)=T(y)T(t).
Proof. [Proof of (eq:reflection)]
-Fix generic $y$. Sample the identity at the special values $t=y,-y,0,4L,
-y+2L,y-2L,y+b,y-b$. Each check reduces to one of (eq:K-swap-trig),
-(eq:K-fusion-below), (eq:K-unitary), $R(0)=P$,
-$R(b)=ΩbarbarΩ^T$Ω^T$ or $R(2L)=WZ$. For instance at $t=y$ both sides
-are $P D_{e,2}(y)^2 P$ by $R(0)=P$ and $D_e(y)D_e(-y)=I$. At $t=y+2L$
-the right side becomes $WD_e((t+y)/2)W^T$ via (eq:K-fusion-below);
-the left side matches by unitarity. At $t=y+b$: right side is
-$ΩΩ^T$ using $D_{e,2}(y)R(t+y)K_e(t) = PD_{e,1}(y)K_e(-y)=Ω$.
+Fix generic y. Sample the identity at the special values t=y,-y,0,4L,
+y+2L,y-2L,y+b,y-b. Each check reduces to one of (eq:K-swap-trig),
+(eq:K-fusion-below), (eq:K-unitary), R(0)=P,
+R(b)=ΩbarbarΩ^TΩ^T or R(2L)=WZ. For instance at t=y both sides
+are P D_{e,2}(y)^2 P by R(0)=P and D_e(y)D_e(-y)=I. At t=y+2L
+the right side becomes WD_e((t+y)/2)W^T via (eq:K-fusion-below);
+the left side matches by unitarity. At t=y+b: right side is
+ΩΩ^T using D_{e,2}(y)R(t+y)K_e(t) = PD_{e,1}(y)K_e(-y)=Ω.
After clearing denominators, the difference
-in (eq:reflection) is a trigonometric polynomial of degree $le6$
-in $t$, periodic/antiperiodic by $pi$. Seven samples suffice.
+in (eq:reflection) is a trigonometric polynomial of degree le6
+in t, periodic/antiperiodic by pi. Seven samples suffice.
The fusion identity for rows.:
@@ -773,24 +772,24 @@ The key algebraic step in extracting the physical eigenvalue is the
following.
[Row fusion]
-For all $t$ and all $iin{1,...,N}$,
+For all t and all iin{1,...,N},
T(t-L) T(t+L) = T(t)quadon the physical sector.
-More precisely, $T(t)^m_{∅,∅}=1$ for all $mge0$
-and all $t$, and the above identity holds on the total-spin-zero
-eigenline of $T(t)$.
+More precisely, T(t)^m_{∅,∅}=1 for all mge0
+and all t, and the above identity holds on the total-spin-zero
+eigenline of T(t).
Proof.
-Use labels $B,C,A,D'$ with parameters $t-L, r+L, t+L, r-L$. Swap the
-middle rows by $R_{CA}(r-t)$, and use (eq:K-fusion-below)
-componentwise; the fused boundary columns from $K_{e_ell}K_{e_ell}$
-become a single $K_{e_ell}$ on the surviving total-spin sector, and
-similarly on the right. The remaining bulk $R$'s act on the two
+Use labels B,C,A,D' with parameters t-L, r+L, t+L, r-L. Swap the
+middle rows by R_{CA}(r-t), and use (eq:K-fusion-below)
+componentwise; the fused boundary columns from K_{e_ell}K_{e_ell}
+become a single K_{e_ell} on the surviving total-spin sector, and
+similarly on the right. The remaining bulk R's act on the two
intermediate fused spins by the fusion equation (eq:R-fusion)
-applied with $u=t$. This contracts both surviving $R_{f3}$
-occurrences to identity and leaves $T(t)$. The details parallel the
+applied with u=t. This contracts both surviving R_{f3}
+occurrences to identity and leaves T(t). The details parallel the
commutation proof, using (eq:K-fusion-below) where the swap proof
used (eq:K-swap-trig).
@@ -800,129 +799,126 @@ used (eq:K-swap-trig).
The physical eigenvalue at homogeneous rapidities.:
-We now pass from the algebraic structure of $T(t)$ to the physical
+We now pass from the algebraic structure of T(t) to the physical
bridge mass. The key step (this subsection) is showing that, at the
-physical specialization $t=gamma$, $xi_j=-L/2$ (all $N$
-sites), the eigenvalue $1$ of $T(gamma)$ corresponds to the same
-planar-path measure that defines $B_N$, and that the mixed coefficient
-in the two boundary variables gives exactly $B_N$.
+physical specialization t=gamma, xi_j=-L/2 (all N
+sites), the eigenvalue 1 of T(gamma) corresponds to the same
+planar-path measure that defines B_N, and that the mixed coefficient
+in the two boundary variables gives exactly B_N.
Planar path expansion.:
-The product $T(gamma)^m$ of $m$ double rows, evaluated between empty
+The product T(gamma)^m of m double rows, evaluated between empty
tensor states on sites and trivial vectors on boundaries, expands as a
sum over planar path systems in an oriented square diagram.
Each local tile (one triangle of the honeycomb) contributes one of the
-following arc configurations, with weights from $a,h,v,c,d$:
+following arc configurations, with weights from a,h,v,c,d:
{c|c}
config & weight
-hline
-empty & $1$
-single e-n or s-w & $a(u)$
-single e-s or n-w & $h(u)$
-single straight-across & $v(u)$
-double (e-n and s-w) & $c(u)$
-double (e-s and n-w) & $d(u)$
-
-
-The arcs carry phases: all $+1$ except e-to-s (orientation from east wall
-to south wall) which gets $s$, and n-to-w which gets $s^{-1}$; reversing
-an arc conjugates. Boundary U-turns are $k(t)l(t)$-weighted at the two
+
+empty & 1
+single e-n or s-w & a(u)
+single e-s or n-w & h(u)
+single straight-across & v(u)
+double (e-n and s-w) & c(u)
+double (e-s and n-w) & d(u)
+
+
+The arcs carry phases: all +1 except e-to-s (orientation from east wall
+to south wall) which gets s, and n-to-w which gets s^{-1}; reversing
+an arc conjugates. Boundary U-turns are k(t)l(t)-weighted at the two
edges.
Loop cancellation.: Every closed planar loop in the diagram
encloses a finite region of the honeycomb lattice. The two possible
orientations of a simple closed loop have tangent-turn angles
-$pm 2pi$ (contractible) or $0$ (essential, wrapping around the
+pm 2pi (contractible) or 0 (essential, wrapping around the
period). Summing the two orientations:
+ contractible: the phases differ by s^{2}-s^{-2}=0;
+ essential: same total phase, so also cancels;
+ at the boundary, U-turns pair off.
-
-item contractible: the phases differ by $s^{2}-s^{-2}=0$;
-item essential: same total phase, so also cancels;
-item at the boundary, U-turns pair off.
-
-Hence the entries of $T(t)$ at $e_ell=e_r=0$ reduce to sums over
+Hence the entries of T(t) at e_ell=e_r=0 reduce to sums over
systems of non-self-intersecting arcs and bridges. In particular
-$T(t)^m_{∅,∅}=1$ for any $t$ (no closed-path
+T(t)^m_{∅,∅}=1 for any t (no closed-path
contribution).
Physical tile weights.:
-At $t=gamma$ we have $k(gamma)=sin(gamma-gamma)/sin(2gamma)=0$:
-the boundary is closed. The sites sit at $xi_j=-L/2$ so the tile
-argument is $gamma+xi_j=L$. Substituting:
+At t=gamma we have k(gamma)=sin(gamma-gamma)/sin(2gamma)=0:
+the boundary is closed. The sites sit at xi_j=-L/2 so the tile
+argument is gamma+xi_j=L. Substituting:
a(L)&=(Ssin(2L))/(sin(3L))=frac{1}{√(2+sqrt2)}=kappa,
h(L)&=v(L)=c(L)=(Ssin(L))/(sin(3L))=(1)/(2+sqrt2)=kappa^2,
d(L)&=(sin(L)sin(0))/(sin(3L))=0.
These are exactly the honeycomb path weights: passing through a
-triangle centre costs $kappa$, and pairing two arcs at a shared centre
-costs $kappa^2=kappacdotkappa$.
-
-The $mto∞$ limit.:
-Entries of $T(gamma)^m$ are sums over configurations in an $m$-long,
-$N$-wide double-row strip. Configurations whose arcs all stay within
-the interior of the strip contribute a factor $le C^{m}$ with $C<1$
-(chosen from the "no-growth" option, weight $1$, per row) and hence
-vanish as $mto∞$. Surviving configurations are unions of
-
-item U-turn boundary arcs at the left and right edges (bounded mass);
-item bridges crossing the strip from left to right (mass by Theorem
-thm:finite-bridges, order $m^{-1/4}$ per unit width).
+triangle centre costs kappa, and pairing two arcs at a shared centre
+costs kappa^2=kappacdotkappa.
+
+The mto∞ limit.:
+Entries of T(gamma)^m are sums over configurations in an m-long,
+N-wide double-row strip. Configurations whose arcs all stay within
+the interior of the strip contribute a factor le C^{m} with C<1
+(chosen from the "no-growth" option, weight 1, per row) and hence
+vanish as mto∞. Surviving configurations are unions of
+ U-turn boundary arcs at the left and right edges (bounded mass);
+ bridges crossing the strip from left to right (mass by Theorem
+thm:finite-bridges, order m^{-1/4} per unit width).
Bridge contributions to the empty-to-empty entry are
phases that cancel by the parafermionic identity (the
Duminil-Copin--Smirnov observable; see the proof of the analogous
-identity in Section~sec:sewing). So the $mto∞$ limiting
+identity in Section~sec:sewing). So the mto∞ limiting
empty-to-empty entry of the normalized projection
-$P_0 := lim_{mto∞}T(gamma)^m$ is
+P_0 := lim_{mto∞}T(gamma)^m is
(P_0)_{∅,∅}=1.
-Simplicity of the eigenvalue $1$.:
-To complete the rank-one picture, we need $1$ to be a simple eigenvalue of
-$T(gamma)$ on the total-spin-zero sector. The proof is by sending one
-site's rapidity to imaginary infinity: the local $R$-matrices become
+Simplicity of the eigenvalue 1.:
+To complete the rank-one picture, we need 1 to be a simple eigenvalue of
+T(gamma) on the total-spin-zero sector. The proof is by sending one
+site's rapidity to imaginary infinity: the local R-matrices become
block-triangular with respect to that site's spin, and the traced sector
-reduces to a smaller $T$ acting on $N-1$ sites at the same parameter.
-By induction on $N$, plus the base case $N=1$ (which is one-site, and
-where the diagonal $R$'s give $1+xi s^{-6p}+xi^{-1}s^{6p}neq 1$),
-simplicity persists at every $N$.
+reduces to a smaller T acting on N-1 sites at the same parameter.
+By induction on N, plus the base case N=1 (which is one-site, and
+where the diagonal R's give 1+xi s^{-6p}+xi^{-1}s^{6p}neq 1),
+simplicity persists at every N.
-Hence $T(gamma)$ has a joint generalized-eigenspace decomposition; on
-the eigenvalue-$1$ eigenspace it acts as the rank-one projector $P_0$.
+Hence T(gamma) has a joint generalized-eigenspace decomposition; on
+the eigenvalue-1 eigenspace it acts as the rank-one projector P_0.
-The mixed coefficient and $q(t;xi)$.:
-Now let $e_ell,e_r$ be formal variables (with $e^{-2}=0$). The
-eigenvalue of $T(t)$ near the physical point, expanded to first order
-in $e_ell e_r$, is
+The mixed coefficient and q(t;xi).:
+Now let e_ell,e_r be formal variables (with e^{-2}=0). The
+eigenvalue of T(t) near the physical point, expanded to first order
+in e_ell e_r, is
-T(t) on the eigenvalue-$1$ linequad↝quad
+T(t) on the eigenvalue-1 linequad↝
1+e_ell,e_r,q(t;xi)+O((e_ell e_r)^2)
-where $q(t;xi)$ is a scalar function. Equivalently, in the mixed
-coefficient of $T(t)^m$,
+where q(t;xi) is a scalar function. Equivalently, in the mixed
+coefficient of T(t)^m,
[e_ell e_r](T(t)^m)_{∅,∅}
= m,q(t;xi)+O(1).
-(The $O(1)$ comes from derivatives of the projection and decays
-exponentially in $m$ by the spectral gap to the next eigenvalue.)
+(The O(1) comes from derivatives of the projection and decays
+exponentially in m by the spectral gap to the next eigenvalue.)
-The coefficient $q$ is the bridge mass.:
-At $t=gamma$ and $xi=-L/2$ (homogeneous), the coefficient $q$
+The coefficient q is the bridge mass.:
+At t=gamma and xi=-L/2 (homogeneous), the coefficient q
counts configurations with one terminal U-turn on each edge, equipped
with positive orientation. These are precisely honeycomb
-bridges: paths entering at a lower terminal, crossing the $2m$-row
-strip, and exiting at an upper terminal. The factor $2$ in
+bridges: paths entering at a lower terminal, crossing the 2m-row
+strip, and exiting at an upper terminal. The factor 2 in
B_N = (1)/(2)q(gamma;xi)
counts the two identical mechanisms (two rows) per bridge.
-Equivalently.: Extraction of $[e_ell e_r]$ in
+Equivalently.: Extraction of [e_ell e_r] in
(eq:m-power), translation invariance, and the bridge count give
B_N = lim_{mto∞}(1)/(2m)
@@ -930,46 +926,46 @@ B_N = lim_{mto∞}(1)/(2m)
=(1)/(2)q(gamma;xi).
-Analytic structure of $q(t;xi)$.:
+Analytic structure of q(t;xi).:
-We now show that $q(t;xi)$, viewed as a function of $t$, has
+We now show that q(t;xi), viewed as a function of t, has
enough structure to be pinned down by finitely many linear conditions.
-Denote $mathbf X(t)=bfX(t)=cos(2t)$.
+Denote mathbf X(t)=bfX(t)=cos(2t).
-[Rational structure of $q$]
-There exists a fixed scalar $rho$ (depending only on $a_*,...$)
-and a polynomial $p(t)$ of $mathbf X(t)$bfX(t)$ of degree at most $N+1$
+[Rational structure of q]
+There exists a fixed scalar rho (depending only on a_*,...)
+and a polynomial p(t) of mathbf X(t)bfX(t) of degree at most N+1
such that
q(t;xi)=frac{p(t)}{(mathbf X+1)^bfX+1)^{N}(mathbf X+bfX+cos 2L)^{N+1}},
-and the polynomial part of $q$ as $mathbf XbfXto∞$ is $4mathbf X+bfX+rho$.
+and the polynomial part of q as mathbf XbfXto∞ is 4mathbf X+bfX+rho.
Proof.
-The $mto∞$ limit and (1) column-by-column: the only pole sources
-in $T(t)$ as a function of $t$ are the zeros of $Q(u)$. For each $U$
-(row transfer), we have $N$ local $R$-matrices plus two boundary
-$K$'s; setting $mathbf X=bfX=cos(2t)$ converts all $t$-dependences to
-rational in $mathbf X$bfX$ since $sin(2t)$ etc. are polynomial in
-$cos(2t)$. The two boundary rows give poles at $mathbf X=-1$bfX=-1$ and
-$mathbf X=-bfX=-cos(2L)$; the $N$ local poles pile at
-$(mathbf X+1)^N(bfX+1)^N(mathbf X+bfX+cos 2L)^{N+1}$. (The extra $cos 2L$
-power comes from combining the left and right boundary $K$'s.)
-
-For the polynomial part: as $mathbf XbfXto∞$, $T(t)$ tends to
-$I+$ single-$t$-matrix terms by unitarity, contributing $4mathbf X$bfX$
-from the leading coefficient; a fixed scalar $rho$ traces the linear
+The mto∞ limit and (1) column-by-column: the only pole sources
+in T(t) as a function of t are the zeros of Q(u). For each U
+(row transfer), we have N local R-matrices plus two boundary
+K's; setting mathbf X=bfX=cos(2t) converts all t-dependences to
+rational in mathbf XbfX since sin(2t) etc. are polynomial in
+cos(2t). The two boundary rows give poles at mathbf X=-1bfX=-1 and
+mathbf X=-bfX=-cos(2L); the N local poles pile at
+(mathbf X+1)^N(bfX+1)^N(mathbf X+bfX+cos 2L)^{N+1}. (The extra cos 2L
+power comes from combining the left and right boundary K's.)
+
+For the polynomial part: as mathbf XbfXto∞, T(t) tends to
+I+ single-t-matrix terms by unitarity, contributing 4mathbf XbfX
+from the leading coefficient; a fixed scalar rho traces the linear
correction.
-[Zeros of $q$]
+[Zeros of q]
The following hold:
q(3L/2;xi)=q(5L/2;xi)=0.
-Moreover, as an identity in the argument $u=t-gamma$,
+Moreover, as an identity in the argument u=t-gamma,
q(gamma+u-L;xi)+q(gamma+u+L;xi)-q(gamma+u;xi)
has a zero of multiplicity
@@ -981,62 +977,61 @@ ge N & at u=L.
Proof.
-First, $q(0)=q(4L)=0$: at $t=0$ and $t=4L$, $T(t)=I$ on the
+First, q(0)=q(4L)=0: at t=0 and t=4L, T(t)=I on the
physical sector (Proposition~prop:T-structure(3)), so the
-eigenvalue $1$ is exact and the $e_ell e_r$ coefficient vanishes.
-Using $T(t)=T(b-t)$ from Proposition~prop:T-structure(2), we
-have $q(b)=q(0)=0$ as well. Passing to $p(u)=q(gamma+u)$ and
-using $b=3L$, $gamma=3L/2$:
+eigenvalue 1 is exact and the e_ell e_r coefficient vanishes.
+Using T(t)=T(b-t) from Proposition~prop:T-structure(2), we
+have q(b)=q(0)=0 as well. Passing to p(u)=q(gamma+u) and
+using b=3L, gamma=3L/2:
[
-p(3L/2)=q(3L/2+gamma)=q(3L)=q(b)=0,qquad
+p(3L/2)=q(3L/2+gamma)=q(3L)=q(b)=0,
p(5L/2)=q(5L/2+gamma)=q(5L/2+3L/2)=q(4L)=0.
]
This proves (eq:q-zeros-1).
For (eq:q-zeros-2): consider
-$T(t-L)T(t+L)-T(t)$ acting on the eigenspace. Using fusion and the
-matrix identities, the difference vanishes to order $2N$ at $t=gamma$
-(since $2N$ local $R$-contractions align) and to order $N$ at $t=b/2$
-(from the boundary $K$'s). Taking mixed coefficients gives
+T(t-L)T(t+L)-T(t) acting on the eigenspace. Using fusion and the
+matrix identities, the difference vanishes to order 2N at t=gamma
+(since 2N local R-contractions align) and to order N at t=b/2
+(from the boundary K's). Taking mixed coefficients gives
(eq:q-zeros-2).
The scalar crossing function.:
-At homogeneous $xi_j$ (i.e., at the physical rapidities with correction
-$-L/2$ already absorbed), define
+At homogeneous xi_j (i.e., at the physical rapidities with correction
+-L/2 already absorbed), define
-p(u)=q(gamma+u;xi),qquad mathbf X(u)=bfX(u)=cos(2(gamma+u)).
+p(u)=q(gamma+u;xi), mathbf X(u)=bfX(u)=cos(2(gamma+u)).
-Then $p$ is even: $p(u)=p(-u)$ (from $T(t)=T(r)$, $r=b-t$,
-so $p(u)=q(gamma+u)=q(b-gamma-u)=q(gamma-u)=p(-u)$). And $p$
-is $pi$-periodic since $T$ is. So $p$ is a function of $mathbf X$,bfX$, and
+Then p is even: p(u)=p(-u) (from T(t)=T(r), r=b-t,
+so p(u)=q(gamma+u)=q(b-gamma-u)=q(gamma-u)=p(-u)). And p
+is pi-periodic since T is. So p is a function of mathbf X,bfX, and
(eq:q-rational) with a change of variable gives
p(u)=frac{P(mathbf X(u))}bfX(u)){(mathbf X+1)^N(bfX+1)^N(mathbf X+bfX+cos 2L)^{N+1}}.
The conditions (eq:q-zeros-1), (eq:q-zeros-2) translate to:
-
-item $P(mathbf X(3L/2))=P(bfX(3L/2))=P(mathbf X(5L/2))=0$,bfX(5L/2))=0$, from
-$p(3L/2)=p(5L/2)=0$ via (eq:q-zeros-1).
-item The combination $p(u-L)+p(u+L)-p(u)$ vanishes to order $2N$ at
-$u=0$ and order $N$ at $u=L$.
+ P(mathbf X(3L/2))=P(bfX(3L/2))=P(mathbf X(5L/2))=0,bfX(5L/2))=0, from
+p(3L/2)=p(5L/2)=0 via (eq:q-zeros-1).
+ The combination p(u-L)+p(u+L)-p(u) vanishes to order 2N at
+u=0 and order N at u=L.
We will use the following functional-analytic representation in the
-next section. For $u$ in the horizontal strip $|Re u|le 5L/2$,
+next section. For u in the horizontal strip |Re u|le 5L/2,
p(u)=mathcal LcalL!((C_u)/(D)),
where
[
-C_u(x)=cosh(usqrt x),quad D(x)=2C_L(x)-1,
+C_u(x)=cosh(usqrt x), D(x)=2C_L(x)-1,
]
-$mathcal L$calL$ is a fixed linear functional, and $x=z^2$.
+mathcal LcalL is a fixed linear functional, and x=z^2.
-The value $2B_N=2cdot(1)/(2)q(gamma)=q(gamma)=p(0)$ becomes
+The value 2B_N=2cdot(1)/(2)q(gamma)=q(gamma)=p(0) becomes
2B_N=p(0)=mathcal L(1/D).calL(1/D).
@@ -1044,45 +1039,45 @@ The value $2B_N=2cdot(1)/(2)q(gamma)=q(gamma)=p(0)$ becomes
The finite functional.:
-We now pin down $mathcal L$.calL$. Set $mu_0(x)=C_{L/2}(x)$,
-$mu_1(x)=C_{3L/2}(x)$, and
-$M=span{1,x,...,x^{N-1};∂_m^j C_m(x)|_{m=L}}_{j=0}^{N-1}$.
+We now pin down mathcal L.calL. Set mu_0(x)=C_{L/2}(x),
+mu_1(x)=C_{3L/2}(x), and
+M=span{1,x,...,x^{N-1};∂_m^j C_m(x)|_{m=L}}_{j=0}^{N-1}.
Then:
[Finite functional]
-There is a unique linear functional $mathcal L$calL$ on the span of
-$M$, $C_{L/2}$, $C_{3L/2}/D$ such that:
+There is a unique linear functional mathcal LcalL on the span of
+M, C_{L/2}, C_{3L/2}/D such that:
mathcal L(f)calL(f) &= 0 for every fin M,
mathcal L(C_calL(C_{L/2}) &= 0,
mathcal L(C_calL(C_{3L/2}-C_{L/2}/D) &= 0.
-For this functional, $2B_N=mathcal L(1/D)$calL(1/D)$ for every $N$.
+For this functional, 2B_N=mathcal L(1/D)calL(1/D) for every N.
Proof. [Outline]
-Apply partial fractions to $p(u)$ in $mathbf X$.bfX$. Standard identities
+Apply partial fractions to p(u) in mathbf X.bfX. Standard identities
int_0^∞(cosh(uz)sinh(mz))/(sinh(pi z/2)),dz
=(1)/(2)[tan(m+u)+tan(m-u)]=(Ssin(2m))/(mathbf X+bfX+cos 2m)
-(at $z$ real, $x=z^2$) give the interpolation basis ${1,x,...,x^{N-1},
-∂_m^j C_m(x)|_{m=L}}_j$. Conditions (eq:q-zeros-2)
+(at z real, x=z^2) give the interpolation basis {1,x,...,x^{N-1},
+∂_m^j C_m(x)|_{m=L}}_j. Conditions (eq:q-zeros-2)
of Section~sec:strip-q-structure translate to
(eq:L-M); (eq:q-zeros-1) translate to
(eq:L-C-L-2) and (eq:L-C-3L-2). The remaining free
-coefficient of $1/D$ is $mathcal L(1/D)=p(0)=2B_N$.calL(1/D)=p(0)=2B_N$.
+coefficient of 1/D is mathcal L(1/D)=p(0)=2B_N.calL(1/D)=p(0)=2B_N.
-The interpolation basis ${1,x,...,x^{N-1}}cup
-{∂_m^j C_m(x)|_{m=L}}_{j=0}^{N-1}$ is $2N$-dimensional,
+The interpolation basis {1,x,...,x^{N-1}}cup
+{∂_m^j C_m(x)|_{m=L}}_{j=0}^{N-1} is 2N-dimensional,
matching the space of rational functions with pole structure
-$(mathbf X+1)^bfX+1)^{-N}(mathbf X+bfX+cos 2L)^{-(N+1)}$ and polynomial
-degree $le 2N+1$ subject to the two homogeneous constraints
+(mathbf X+1)^bfX+1)^{-N}(mathbf X+bfX+cos 2L)^{-(N+1)} and polynomial
+degree le 2N+1 subject to the two homogeneous constraints
(eq:q-zeros-1) and (eq:q-zeros-2). The identity
(eq:cheb-integral) is a standard residue computation:
-contour shift to $z=2ik$, $kge1$, picks up the simple residues of
-$sinh(pi z/2)^{-1}$ and produces a Fourier series in $u$ that
-sums to the displayed $tan$-combination.
+contour shift to z=2ik, kge1, picks up the simple residues of
+sinh(pi z/2)^{-1} and produces a Fourier series in u that
+sums to the displayed tan-combination.
@@ -1090,32 +1085,31 @@ sums to the displayed $tan$-combination.
The positive interpolation estimate.:
Section~c6:strip-transfer reduced the strip bridge mass to a
-single linear functional $mathcal L$calL$ on a $(2N+2)$-dimensional space.
-Here we evaluate $mathcal L(1/D)$calL(1/D)$ up to an absolute constant, obtaining
-$B_Nasymp N^{-1/4}$. The engine is a determinant-density representation
+single linear functional mathcal LcalL on a (2N+2)-dimensional space.
+Here we evaluate mathcal L(1/D)calL(1/D) up to an absolute constant, obtaining
+B_Nasymp N^{-1/4}. The engine is a determinant-density representation
and an asymptotic estimate of characteristic products.
Spaces, measure, determinant density.:
Retain from Proposition~prop:finite-functional:
-
-item measure $dmu(x)=zD(x),dz/sinh(4Lz)$, $x=z^2$;
-item $M_N=span{1,x,...,x^{N-1};∂_m^jC_m(x)|_{m=L}}_{j=0}^{N-1}$;
-item $mathcal L(M_N)=calL(M_N)=mathcal L(C_calL(C_{L/2})=
-mathcal L(C_calL(C_{3L/2}-C_{L/2}/D)=0$;
-item $2B_N=mathcal L(1/D)$.calL(1/D)$.
+ measure dmu(x)=zD(x),dz/sinh(4Lz), x=z^2;
+ M_N=span{1,x,...,x^{N-1};∂_m^jC_m(x)|_{m=L}}_{j=0}^{N-1};
+ mathcal L(M_N)=calL(M_N)=mathcal L(C_calL(C_{L/2})=
+mathcal L(C_calL(C_{3L/2}-C_{L/2}/D)=0;
+ 2B_N=mathcal L(1/D).calL(1/D).
Set
-c_1&=C_{L/2},quad c_3=C_{3L/2},quad
-h_0=1/D,quad h_1=c_1/D,quad g(x)=sinh(Lsqrt x)/sqrt x,
-F_0&=M_N+mathbb Cc_1,bbCc_1,quad F=F_0+mathbb Cc_3,bbCc_3,quad G=M_N+mathbb Cg.bbCg.
+c_1&=C_{L/2}, c_3=C_{3L/2},
+h_0=1/D, h_1=c_1/D, g(x)=sinh(Lsqrt x)/sqrt x,
+F_0&=M_N+mathbb Cc_1,bbCc_1, F=F_0+mathbb Cc_3,bbCc_3, G=M_N+mathbb Cg.bbCg.
-Note $g(x)=int_0^L C_t(x),dt$.
+Note g(x)=int_0^L C_t(x),dt.
-Let $d=2N+1=dim F_0$. For ordered $X=(x_1<...<x_d)$ in $(0,∞)^d$,
-let $det F_0(X)$, $det G(X)$ denote evaluation determinants (rows =
+Let d=2N+1=dim F_0. For ordered X=(x_1<...<x_d) in (0,∞)^d,
+let det F_0(X), det G(X) denote evaluation determinants (rows =
basis functions, columns = points, jet values at coincident points).
Put
@@ -1125,170 +1119,170 @@ nu = frac{det F_0(X)det G(X)prod_{i=1}^d dmu(x_i)}
By Lemma~lem:jet-positivity below this is a probability density.
[Jet positivity]
-Enumerate "labels" $betain{0,L^2}$ each with multiplicity $N$, plus
-one $g$-row inserted between the two label blocks. For any ordered
-$X_1<...<X_d$ in $(0,∞)$, every jet-determinant formed from
-complete jets of $C_{sqrtbeta}(x)$ at the labels is positive; at
-confluent points it is nonnegative. In particular $M_N$, $F_0$, $F$, $G$
-are Chebyshev on $[0,∞)$, counting multiplicities.
+Enumerate "labels" betain{0,L^2} each with multiplicity N, plus
+one g-row inserted between the two label blocks. For any ordered
+X_1<...<X_d in (0,∞), every jet-determinant formed from
+complete jets of C_{sqrtbeta}(x) at the labels is positive; at
+confluent points it is nonnegative. In particular M_N, F_0, F, G
+are Chebyshev on [0,∞), counting multiplicities.
Proof.
-$C_{sqrtbeta}(x)=sum_{jge0}beta^jx^j/(2j)!$ has nonnegative
+C_{sqrtbeta}(x)=sum_{jge0}beta^jx^j/(2j)! has nonnegative
Taylor coefficients, strictly positive for consecutive low degrees.
Ordered power determinants on positive nodes are positive by Rolle and
coalescing; confluent cases are nonnegative limits. Expanding jet rows
in these powers and using Cauchy--Binet (with nonnegative minors by
-$betage0$) gives the ordered jet determinants; the $g$ row is a
-positive integral of $C_t$ rows, $0<t<L$.
+betage0) gives the ordered jet determinants; the g row is a
+positive integral of C_t rows, 0<t<L.
[Determinant density]
-For $H$ with $int|Hf|dmu<∞$ ($fin M_N+mathbb Cg$):bbCg$):
+For H with int|Hf|dmu<∞ (fin M_N+mathbb Cg):bbCg):
(mathcal L(H))/(4a_*)=calL(H))/(4a_*)=nu[e_{-4}H]+s_*,nu[e_0H],
-with $e_{-a}H=H(-a)-I_{F_0,X}H(-a)$ the residual of $F_0$-interpolation
-at nodes $X$, and $s_*=rho/(4a_*)$ the scalar in $mathcal L$.calL$.
+with e_{-a}H=H(-a)-I_{F_0,X}H(-a) the residual of F_0-interpolation
+at nodes X, and s_*=rho/(4a_*) the scalar in mathcal L.calL.
Proof.
Apply Andr'eief's identity
-$intdet F_0det Gprod dmu=det(int F_0G^Tdmu)$ row by row: the
-integral $int Hf,dmu$ against the density reduces to a projective
-$nu$-average; point evaluations at $-4$ and $0$ enter as residual
-evaluation vectors because $H$ is fixed on the discrete $mathcal L$calL$
+intdet F_0det Gprod dmu=det(int F_0G^Tdmu) row by row: the
+integral int Hf,dmu against the density reduces to a projective
+nu-average; point evaluations at -4 and 0 enter as residual
+evaluation vectors because H is fixed on the discrete mathcal LcalL
component. See Tracy--Widom [cite] for the identity
itself.
-We will use the Schur complement identity on the appended rows $c_3,
-h_0$ (eliminating $c_3$ from the $F_0$ determinant):
+We will use the Schur complement identity on the appended rows c_3,
+h_0 (eliminating c_3 from the F_0 determinant):
nu[e_0c_3],nu[e_{-4}h_0]-nu[e_{-4}c_3],nu[e_0h_0]
=nu![(e_0c_3)E_{-4}h_0],
-where $E_y$ is the residual with row $c_3$ appended before the
-$h_0$-residual.
+where E_y is the residual with row c_3 appended before the
+h_0-residual.
Also define the tilted measure
dnu_3(X)propto -e_0c_3(X),dnu(X).
-Since $-e_0c_3>0$ by jet positivity ($d$ is odd), $nu_3$ is a
+Since -e_0c_3>0 by jet positivity (d is odd), nu_3 is a
probability density.
Characteristic products.:
[Characteristic products]
-Fix $0<y<z$ and $r(x)=(x+y^2)/(x+z^2)$. For either $eta=nu$ or
-$eta=nu_3$ and any $N$-subset of the coordinates,
+Fix 0<y<z and r(x)=(x+y^2)/(x+z^2). For either eta=nu or
+eta=nu_3 and any N-subset of the coordinates,
eta![prod_{i=1}^Nr(x_i)]asymp N^{b(y-z)/pi}.
-In the growing regime $y^2=2z^2=2N^{3/2}$ (where $1le rle2$)
-the same expectation is at most $exp(o(N))$.
+In the growing regime y^2=2z^2=2N^{3/2} (where 1le rle2)
+the same expectation is at most exp(o(N)).
Proof. [Sketch]
-Interlacing. Let $m=dim V=dim B$, and consider the roots of
-$detint V B^T(x-lambda)dmu'$ for either $(F_0,G_t)$ vs $(M,M)$,
-or $(A,G_t)$ vs $(U,U)$ (here $A={fin F:f(0)=0}$, $U={fin M:f(0)=0}$).
+Interlacing. Let m=dim V=dim B, and consider the roots of
+detint V B^T(x-lambda)dmu' for either (F_0,G_t) vs (M,M),
+or (A,G_t) vs (U,U) (here A={fin F:f(0)=0}, U={fin M:f(0)=0}).
The roots of the smaller pair interlace those of the larger. This is
the Chebyshev sign-change argument: any nonzero real analytic function
-orthogonal to a dimension-$m$ Chebyshev space flips sign at least $m$
+orthogonal to a dimension-m Chebyshev space flips sign at least m
times; consecutive interlacing subspaces cannot share a positive root.
-Deform all $beta$-labels to 0 continuously; interlacing is preserved.
+Deform all beta-labels to 0 continuously; interlacing is preserved.
-Reduction to polynomial. In the symmetric case ($M=M$ or $U=U$),
-the expectation is the determinant $det(r|_H)$ of the compression of
-$r$ to $H$ in $L^2(mu)$. By $L^2(mu)$ approximation, $H$ lies between
-the polynomial spaces $mathcal P_calP_{<N}$ and $mathcal P_calP_{<CN}$ for
-a large absolute $C$, with operator-norm error $e^{-N}$. Hence the
-determinant changes by $1+o(1)$.
+Reduction to polynomial. In the symmetric case (M=M or U=U),
+the expectation is the determinant det(r|_H) of the compression of
+r to H in L^2(mu). By L^2(mu) approximation, H lies between
+the polynomial spaces mathcal P_calP_{<N} and mathcal P_calP_{<CN} for
+a large absolute C, with operator-norm error e^{-N}. Hence the
+determinant changes by 1+o(1).
-Polynomial determinants. For $mu_c$ the probability measure
-$dp/cosh(cp)$ on $p=sqrt x$, put $alpha=pi/(2c)$ and
+Polynomial determinants. For mu_c the probability measure
+dp/cosh(cp) on p=sqrt x, put alpha=pi/(2c) and
-A_m(x)=cos(alpha m)C_m(x),quad
-int A_mA_n,dmu_c=(1)/(1-w_mw_n),quad w_m=tan(alpha m).
+A_m(x)=cos(alpha m)C_m(x),
+int A_mA_n,dmu_c=(1)/(1-w_mw_n), w_m=tan(alpha m).
(The identity follows from
-$int C_mC_ndmu_c=frac12(secalpha(m+n)+secalpha(m-n))$, itself a
+int C_mC_ndmu_c=frac12(secalpha(m+n)+secalpha(m-n)), itself a
direct Fourier transform.) This gives explicit orthogonal polynomials
-$P_j$ and permits precise evaluation:
-$K_{n,mu_c}(-v^2)asymp n^{2cv/pi}$ for fixed $v>0$, where
-$K_{n,W'}(v)=sup_{deg hle n}|h(v)|^2/int|h|^2dW'$ is the
+P_j and permits precise evaluation:
+K_{n,mu_c}(-v^2)asymp n^{2cv/pi} for fixed v>0, where
+K_{n,W'}(v)=sup_{deg hle n}|h(v)|^2/int|h|^2dW' is the
normalization. Then
R_{W,n}(r^2)=K_{n,W'}(-y^2)/K_{n,W'}(-z^2)
asymp N^{2b(y-z)/pi}.
-This is the Gram cofactor identity on degrees $le n$ for the
-hyperplanes spanned by $(x+y^2)x^j$ and $(x+z^2)x^j$, $0le j<n$.
+This is the Gram cofactor identity on degrees le n for the
+hyperplanes spanned by (x+y^2)x^j and (x+z^2)x^j, 0le j<n.
-Cauchy--Schwarz. Since $r^2$ is the product of two compressed
-determinants with $r$ and $r'=rmu/mu$, the square-root asymptotic
+Cauchy--Schwarz. Since r^2 is the product of two compressed
+determinants with r and r'=rmu/mu, the square-root asymptotic
(eq:char-products) follows from (eq:poly-det) plus Jensen
-on $log r$.
+on log r.
-The exponentially small probability of ${x_{N-1}<N^{3/2}}$ follows
+The exponentially small probability of {x_{N-1}<N^{3/2}} follows
by applying the growing regime: there the product is at least
-$(3/2)^{N-1}$, forcing its expectation under a probability to be tiny.
+(3/2)^{N-1}, forcing its expectation under a probability to be tiny.
Interpolation residuals.:
[Residual bound]
-Let $lambda_*=pi/(3L)=8/3$. For $Ain{e,E}$ and $ain{0,4}$ and
-$jin{0,1}$, and for $v_i$ the interpolation nodes in increasing order:
+Let lambda_*=pi/(3L)=8/3. For Ain{e,E} and ain{0,4} and
+jin{0,1}, and for v_i the interpolation nodes in increasing order:
0le A_{-a}h_jle Cprod_{i=1}^N(v_i+a)/(v_i+lambda_*^2)
-uniformly in $X$. For $h_0$ the analogous lower bound with
-$prod_{i=1}^d$ also holds.
+uniformly in X. For h_0 the analogous lower bound with
+prod_{i=1}^d also holds.
Proof. [Sketch]
-Translate. Put $w=x+a$, so evaluation is at $0$ with strictly
+Translate. Put w=x+a, so evaluation is at 0 with strictly
positive ordered nodes.
Monodromy. In this coordinate, each residual against the jet
space is a convex combination of monomial residuals with powers
-$0,...,N-1$ and at most one additional power. This uses Cauchy--Binet
+0,...,N-1 and at most one additional power. This uses Cauchy--Binet
in the evaluation and error determinants, where the auxiliary minors
-$D_J(0)$ are positive with derivative ratio bounded by
-$theta_N=(sumbeta_i)/(2(2N+1))$; for large $N$ the minors remain
-positive after translating by $a$.
-
-Laplace structure. Write $h_j(w-a)=h_j(-a)mathbb EbbE e^{-wZ_j}$,
-where $Z_jge0$ is a random variable. Explicitly, the product formula
-for $D(x)=2C_L(x)-1$ (via $cosh$-product) gives
-$D(x)propto(x+lambda_*^2)prod_{kge2}(x+s_k)$ with
-$s_k=((2pi k+pi/3)/L)^2$; similarly $c_1$ and $(c_3-c_1)$ have
+D_J(0) are positive with derivative ratio bounded by
+theta_N=(sumbeta_i)/(2(2N+1)); for large N the minors remain
+positive after translating by a.
+
+Laplace structure. Write h_j(w-a)=h_j(-a)mathbb EbbE e^{-wZ_j},
+where Z_jge0 is a random variable. Explicitly, the product formula
+for D(x)=2C_L(x)-1 (via cosh-product) gives
+D(x)propto(x+lambda_*^2)prod_{kge2}(x+s_k) with
+s_k=((2pi k+pi/3)/L)^2; similarly c_1 and (c_3-c_1) have
product form. Hence
-$mathbb P(Z_jbbP(Z_jge t)asymp e^{-lambda_*^2t}$ (for $h_1$, one pairs
-each numerator factor with a distinct smaller $s_k$, splitting off
+mathbb P(Z_jbbP(Z_jge t)asymp e^{-lambda_*^2t} (for h_1, one pairs
+each numerator factor with a distinct smaller s_k, splitting off
the smallest rate by a change-of-measure uniform bound).
Monotone interpolation. Monomial residuals are monotone in
-vertex powers: the $e^{tw}w^j$ coefficient matrix is totally
-nonnegative (exponential of the upper shift times $t$), so the
-old interpolant's coefficient at each node has sign $(-1)^i$; a
+vertex powers: the e^{tw}w^j coefficient matrix is totally
+nonnegative (exponential of the upper shift times t), so the
+old interpolant's coefficient at each node has sign (-1)^i; a
single-power increase changes the residual by a sign-consistent
amount. Reduce to the pure-exponential case:
[
mathbb E[e^bbE[e^{-(lambda_*^2-a)Y}]=prod_{i}(v_i+a)/(v_i+lambda_*^2)
]
-since $Y$ is a sum of exponential variables with rates $v_i+a$.
+since Y is a sum of exponential variables with rates v_i+a.
-Under the measures of (eq:nu-density), the omitted $O(N)$
-factors in (eq:residual-bd) above $N^{3/2}$ give $e^{o(N^{-1/2})}$;
+Under the measures of (eq:nu-density), the omitted O(N)
+factors in (eq:residual-bd) above N^{3/2} give e^{o(N^{-1/2})};
and the small-node probability is exponentially small. Thus
-nu[-e_0c_3]gtrsim N^{-1/2},quad
-|nu[e_0h_i]|le N^{-1+o(1)},quad
+nu[-e_0c_3]gtrsim N^{-1/2},
+|nu[e_0h_i]|le N^{-1+o(1)},
|nu[e_{-4}h_i]|=O(1),
and the tilted target
@@ -1298,92 +1292,92 @@ nu_3[E_{-4}h_0]asymp N^{-1/4}.
Increasing-weight comparison.:
-The final size estimate on $nu_3[-e_0c_3]$ uses monotone coupling.
+The final size estimate on nu_3[-e_0c_3] uses monotone coupling.
[Increasing-weight comparison]
-Let $mu_bpropto dx/cosh(bsqrt x)$ and $nu_b$ be the determinant
-density built from $mu_b$ and the same Jacobi jet structure. The
-ratio of base measures $mu_b/mu$ (up to constants) is the increasing
+Let mu_bpropto dx/cosh(bsqrt x) and nu_b be the determinant
+density built from mu_b and the same Jacobi jet structure. The
+ratio of base measures mu_b/mu (up to constants) is the increasing
function
-phi(x)=(z(cosh(4z)+cosh(2z)-cosh(3z)))/(sinh(4z)),quad z=Lsqrt x,
+phi(x)=(z(cosh(4z)+cosh(2z)-cosh(3z)))/(sinh(4z)), z=Lsqrt x,
with increasing series-coefficient ratios
-$(2j+1)/(4)(1+(1/4)^j-(9/16)^j)/1$. Therefore, for any bounded
-coordinatewise increasing function $f$ on $(0,∞)^d$:
-$nu_b[f]le Cnu[f]$.
+(2j+1)/(4)(1+(1/4)^j-(9/16)^j)/1. Therefore, for any bounded
+coordinatewise increasing function f on (0,∞)^d:
+nu_b[f]le Cnu[f].
Proof.
-A probability measure $eta$ on $(0,∞)^d$ is $MTP_2$ if its
-density $w$ satisfies $w(xwedge y)w(xvee y)ge w(x)w(y)$ for all
-$x,y$, with $xwedge y$ and $xvee y$ the coordinatewise meet and join.
+A probability measure eta on (0,∞)^d is MTP_2 if its
+density w satisfies w(xwedge y)w(xvee y)ge w(x)w(y) for all
+x,y, with xwedge y and xvee y the coordinatewise meet and join.
-Step 1: both measures are $MTP_2$. For measures whose density
-has the determinant-product form $w=det(F,G)prod_i dmu(x_i)$,
-$MTP_2$ holds whenever the base $mu$ is $MTP_2$; the proof
+Step 1: both measures are MTP_2. For measures whose density
+has the determinant-product form w=det(F,G)prod_i dmu(x_i),
+MTP_2 holds whenever the base mu is MTP_2; the proof
(Bogso cite[Theorem 2.27]{Bogso2014}, following Karlin--McGregor)
-reduces the $MTP_2$ inequality to Pl"ucker's $2times 2$ identity on
-the alternating minors $det(F,G)$ and then to
-$log$-supermodularity of the single-variable base density. Both
-$nu$ and $nu_b$, built over the same jet spaces from the base
-measures $dmu$ and $dmu_b$, are therefore $MTP_2$.
-
-Step 2: FKG. For an $MTP_2$ measure $eta$ and any
-coordinatewise nondecreasing bounded $f_1,...,f_m$,
-$eta[prod_jf_j]geprod_jeta[f_j]$ (Harris). Indeed one-coordinate
+reduces the MTP_2 inequality to Pl"ucker's 2times 2 identity on
+the alternating minors det(F,G) and then to
+log-supermodularity of the single-variable base density. Both
+nu and nu_b, built over the same jet spaces from the base
+measures dmu and dmu_b, are therefore MTP_2.
+
+Step 2: FKG. For an MTP_2 measure eta and any
+coordinatewise nondecreasing bounded f_1,...,f_m,
+eta[prod_jf_j]geprod_jeta[f_j] (Harris). Indeed one-coordinate
heat-bath smoothing preserves the FKG inequality: conditional
densities of one variable given the rest have monotone likelihood
ratio in the other coordinates by Pl"ucker's quadratic relation, so
-the FKG product inequality passes to the full $d$-variable
+the FKG product inequality passes to the full d-variable
distribution; the common unit-quantile coupling then extends the
inequality from indicators of increasing cylinders to all bounded
increasing integrands.
Step 3: the comparison. From the lemma computation,
-$dnu_b/dnu=prod_iphi(x_i)$ with $phi$ the increasing
+dnu_b/dnu=prod_iphi(x_i) with phi the increasing
one-variable ratio displayed above, whose Taylor coefficients of
-$philog z$ (recall $z=Lsqrt x$) are nonnegative. Thus $(nu,
-nu_b)$ form an $MTP_2$ pair with ordered densities. Bogso
+philog z (recall z=Lsqrt x) are nonnegative. Thus (nu,
+nu_b) form an MTP_2 pair with ordered densities. Bogso
cite[Theorem 2.27]{Bogso2014} gives the ordered-density corollary
-$nu_b[f]le Cnu[f]$ for bounded coordinatewise increasing $f$ and
-finite $C$ depending only on the jet structure and the density ratio.
+nu_b[f]le Cnu[f] for bounded coordinatewise increasing f and
+finite C depending only on the jet structure and the density ratio.
This is the asserted comparison; combining with Step 2 gives the
stated constant.
-Under $mu_b$, the interpolant of $A_m$ at 0 is a rational function
-in $w=w_m$ whose denominator has roots $w$-labels of $G_t$;
-the residual at $w=1$ is
+Under mu_b, the interpolant of A_m at 0 is a rational function
+in w=w_m whose denominator has roots w-labels of G_t;
+the residual at w=1 is
(1)/(1+s)prod_i(1+sb_i')/(1-b_i')prod_i(1-a_i')/(s+a_i'),
-where $a_i',b_i'in[0,1/3]$ are the $w$-labels of $F_0$ and $G_t$.
-This is uniformly $gtrsim d^{-1/2}$ on $1<s<1+1/d$, giving
-$nu_b[-e_0c_3]gtrsim d^{-1/2}$.
+where a_i',b_i'in[0,1/3] are the w-labels of F_0 and G_t.
+This is uniformly gtrsim d^{-1/2} on 1<s<1+1/d, giving
+nu_b[-e_0c_3]gtrsim d^{-1/2}.
Assembly.:
Combine (eq:schur-identity), (eq:P-7-estimates), and
Lemma~lem:weight-comp. Write
-$A_y=nu[e_yc_3]$, $H_{j,y}=nu[e_yh_j]$
-($yin{0,-4}$). The constraint $mathcal L(c_3-h_1)=0$calL(c_3-h_1)=0$ determines
-$s_*=-frac{A_{-4}-H_{1,-4}}{A_0-H_{1,0}}$; the denominator is
-nonzero for large $N$ since $|A_0|gtrsim N^{-1/2}$ and
-$|H_{1,0}|le N^{-1+o(1)}$.
+A_y=nu[e_yc_3], H_{j,y}=nu[e_yh_j]
+(yin{0,-4}). The constraint mathcal L(c_3-h_1)=0calL(c_3-h_1)=0 determines
+s_*=-frac{A_{-4}-H_{1,-4}}{A_0-H_{1,0}}; the denominator is
+nonzero for large N since |A_0|gtrsim N^{-1/2} and
+|H_{1,0}|le N^{-1+o(1)}.
-The exact form of $mathcal L(h_0)/(4a_*)$calL(h_0)/(4a_*)$ is then
+The exact form of mathcal L(h_0)/(4a_*)calL(h_0)/(4a_*) is then
(mathcal L(h_0))/(4a_*)calL(h_0))/(4a_*)
=frac{A_0,nu_3[E_{-4}h_0]
+H_{1,-4}H_{0,0}-H_{1,0}H_{0,-4}}
{A_0-H_{1,0}}.
-The numerator has leading term $A_0nu_3[E_{-4}h_0]asymp N^{-1/4}cdot
-N^{-1/2}=N^{-3/4}$; the remaining quotient contributes
-$O(N^{-1/2+o(1)})$, negligible. So
-$mathcal L(h_0)/(4a_*)calL(h_0)/(4a_*)asymp N^{-1/4}$, hence
-$B_Nasymp N^{-1/4}$.
+The numerator has leading term A_0nu_3[E_{-4}h_0]asymp N^{-1/4}cdot
+N^{-1/2}=N^{-3/4}; the remaining quotient contributes
+O(N^{-1/2+o(1)}), negligible. So
+mathcal L(h_0)/(4a_*)calL(h_0)/(4a_*)asymp N^{-1/4}, hence
+B_Nasymp N^{-1/4}.
With this, Theorem~thm:finite-bridges items (eq:bridge-mass)
are proved. The remaining items (first-moment and long-bridge) are
@@ -1396,30 +1390,30 @@ The periodic vacuum and its physical pairing.:
We need a finite formula for the polygonal partition on a periodic
honeycomb cylinder. The formula expresses that partition as a ratio
-$G_N(v)/(2f_N(v)^2)$ of two explicit trigonometric polynomials,
-evaluated at homogeneous spectral variables $v_1=...=v_N=0$. The
+G_N(v)/(2f_N(v)^2) of two explicit trigonometric polynomials,
+evaluated at homogeneous spectral variables v_1=...=v_N=0. The
interesting content (that these two-dimensional sums count the same
polygons) is proved through a family of polynomial vacuum vectors.
The cylinder and the finite formula.:
-Take the lozenge-cylinder of period $N$. Put corners $A,B$ of tiles
-on a common horizontal line, $A$ just before column $1$ and $B$ just
-after column $m$, with $m+n=N$. Let $mathcal Z_N(y)$calZ_N(y)$ sum systems of
+Take the lozenge-cylinder of period N. Put corners A,B of tiles
+on a common horizontal line, A just before column 1 and B just
+after column m, with m+n=N. Let mathcal Z_N(y)calZ_N(y) sum systems of
finitely many disjoint simple polygons in the cylinder, each with
-weight $kappa^{rm length}$ and a factor $y+y^{-1}$. Every polygon
-in a system separates $A$ from $B$; the empty system is included.
+weight kappa^{rm length} and a factor y+y^{-1}. Every polygon
+in a system separates A from B; the empty system is included.
-Introduce spectral variables $v_1,...,v_N$, one per column. Put
-$C=2-sqrt2=4cos^2 b$ and
+Introduce spectral variables v_1,...,v_N, one per column. Put
+C=2-sqrt2=4cos^2 b and
-R_s(d)=(-1)/(sin(d+L)sin(d-L)),quad R_c(d)=cos^2 d,quad
+R_s(d)=(-1)/(sin(d+L)sin(d-L)), R_c(d)=cos^2 d,
D(d)=sin(d+2L)sin(d-2L)sin(d+b)sin(d-b).
-Sum over site labels $alpha_jin{+1,0,-1}$, with total sum in the
-two groups ${1,...,m}$ and ${m+1,...,N}$ equal to a common $h$.
-Weight every configuration by $y^hprod_jc_{alpha_j}$ with
-$c_pm=sqrt C$, $c_0=2$, and by the two-body factors
+Sum over site labels alpha_jin{+1,0,-1}, with total sum in the
+two groups {1,...,m} and {m+1,...,N} equal to a common h.
+Weight every configuration by y^hprod_jc_{alpha_j} with
+c_pm=sqrt C, c_0=2, and by the two-body factors
P_{alphaalpha'}(v_j-v_i)=
1,&(alpha,alpha')in{(+,+),(-,-)},
@@ -1428,9 +1422,9 @@ H(v_j-v_i-L),&(alpha,alpha')in{(+,0),(0,-)},
H(v_j-v_i-L)H(v_j-v_i-2L),&(alpha,alpha')=(+,-),
-with $P_{alphaalpha'}(d)=P_{alpha'alpha}(-d)$, and with $H=R_s$
-for a same-group pair, $H=R_c$ across groups. Same-group pairs are
-additionally multiplied by $D(v_j-v_i)$. The resulting sum is
+with P_{alphaalpha'}(d)=P_{alpha'alpha}(-d), and with H=R_s
+for a same-group pair, H=R_c across groups. Same-group pairs are
+additionally multiplied by D(v_j-v_i). The resulting sum is
G_N(v) = sum_{{alpha_j}, sum_{1}^malpha_j=sum_{m+1}^Nalpha_j}
y^{sumalpha_j}prod_jc_{alpha_j}prod_{i<j}P_{alpha_ialpha_j}(v_j-v_i).
@@ -1438,29 +1432,29 @@ y^{sumalpha_j}prod_jc_{alpha_j}prod_{i<j}P_{alpha_ialpha_j}(v_j-v_i).
The denominator is a Pfaffian. Put
-B(u)=(sin(u+b)sin(u-b))/(sin u),qquad h(u)=(cos u)/(B(u)).
+B(u)=(sin(u+b)sin(u-b))/(sin u), h(u)=(cos u)/(B(u)).
-Let $mathsf h_N$sfh_N$ be the skew matrix with upper entries
-$(mathsf h_N)_sfh_N)_{jk}=h(v_k-v_j)$, $j<k$; for odd $N$ append a last
-column of ones and a row of minus ones with bottom-right entry $0$.
-Set $a_{2l}=2^l/sqrt 2$, $a_{2l+1}=(-2)^l$, and
+Let mathsf h_Nsfh_N be the skew matrix with upper entries
+(mathsf h_N)_sfh_N)_{jk}=h(v_k-v_j), j<k; for odd N append a last
+column of ones and a row of minus ones with bottom-right entry 0.
+Set a_{2l}=2^l/sqrt 2, a_{2l+1}=(-2)^l, and
f_N(v)=a_Nprod_{j<k}B(v_k-v_j),Pf,mathsf h_N.sfh_N.
-Explicitly, $f_0=1/sqrt2$ and $f_1=1$, and for even size
+Explicitly, f_0=1/sqrt2 and f_1=1, and for even size
2f_{2m'}^2=4^{m'}prod_{i<j}B(v_j-v_i)^2,det[h(v_j-v_i)]_{i,j=1}^{2m'}.
The apparent singularities in~(eq:f_N) are removable
-(Lemma~lem:poly-vacuum below); $f_N$ is a trigonometric polynomial.
+(Lemma~lem:poly-vacuum below); f_N is a trigonometric polynomial.
[Homogeneous physical pairing]
-For $y>0$,
+For y>0,
mathcal Z_N(y)=calZ_N(y)=left.(G_N(v))/(2f_N(v)^2)|_{v_1=...=v_N=0}.
The denominator is nonzero there, and the identity holds also as a
-Laurent-polynomial identity in $y$.
+Laurent-polynomial identity in y.
The proof is by polynomial interpolation: the right-hand side is
@@ -1469,156 +1463,154 @@ family of fusion conditions.
Periodic eigenlines.:
-Let $K(xi)_{pp}=xi^p$ denote the spin twist ($pin{+1,0,-1}$)
-and $T_xi(t)$ the single-row transfer matrix
+Let K(xi)_{pp}=xi^p denote the spin twist (pin{+1,0,-1})
+and T_xi(t) the single-row transfer matrix
T_xi(t)=Tr_a,K_a(xi)R_{a1}(t+v_1)... R_{aN}(t+v_N),
qquadxi=i or -i,
acting on total-spin-zero site states. The empty site vector is
-denoted $∅$; site order is site 1 first even though operator
+denoted ∅; site order is site 1 first even though operator
action is from the right.
[Polynomial vacuum]
-For $Nge1$ and generic $(v_j)$, $T_i(t)$ has a simple common eigenline
-with eigenvalue $1$. Its normalized eigenvector $w^i$ has
-$w^i_∅=1$, and $Psi_N:=f_Nw^i$ (with $Psi_0=1/sqrt 2$)
-is a trigonometric polynomial of degree $le N-1$ in each $v_j$,
+For Nge1 and generic (v_j), T_i(t) has a simple common eigenline
+with eigenvalue 1. Its normalized eigenvector w^i has
+w^i_∅=1, and Psi_N:=f_Nw^i (with Psi_0=1/sqrt 2)
+is a trigonometric polynomial of degree le N-1 in each v_j,
with the following properties:
-itemParity. On a component with site spin $p_j$, $Psi_N$ is
-$(-1)^{N-1+p_j}$-parity in $v_jmapsto v_j+pi$.
-itemFusion. With remaining list $U$,
+itemParity. On a component with site spin p_j, Psi_N is
+(-1)^{N-1+p_j}-parity in v_jmapsto v_j+pi.
+itemFusion. With remaining list U,
Psi_N(u,u+2L,U)=prod_{zin U}cos(z-u-L)cdot W_{12}Psi_{N-1}(u+L,U),
Psi_N(u,u+b,U)=2cos bprod_{zin U}cos(z-u-L)cos(z-u-2L)cdotΩ_{12}Psi_{N-2}(U).
-itemInfinity. As $|v_N|to i∞$,
-$Psi_N(v)/prod_{j<N}cos(v_N-v_j)tosqrt2Psi_{N-1}(v_1,...,v_{N-1}) ⊗ |0>_N$.
-itemWheel zero. $Psi_N$ vanishes at the wheel
-$(v_1,v_2,v_3)=(u,u+2L,u+5L)$ (order $ge1$; order $ge2$ for twist $-i$).
+itemInfinity. As |v_N|to i∞,
+Psi_N(v)/prod_{j<N}cos(v_N-v_j)tosqrt2Psi_{N-1}(v_1,...,v_{N-1}) ⊗ |0>_N.
+itemWheel zero. Psi_N vanishes at the wheel
+(v_1,v_2,v_3)=(u,u+2L,u+5L) (order ge1; order ge2 for twist -i).
Proof. [Proof sketch]
-Eigenvalue and simplicity. Let $T_j=T_xi(-v_p+jL)$ at any site $p$.
+Eigenvalue and simplicity. Let T_j=T_xi(-v_p+jL) at any site p.
Yang--Baxter and spin conservation give
-T_0T_2=T_1,quad T_0T_3=I,quad T_1T_3=T_2.
+T_0T_2=T_1, T_0T_3=I, T_1T_3=T_2.
-(Check by applying $R_{2p}(0)R_{1p}(2L)=I$ on the image of $W$.)
-Any finite product at common twist has $∅,∅$ entry
-$1$ (loop cancellation, as in Section~sec:strip-physical).
+(Check by applying R_{2p}(0)R_{1p}(2L)=I on the image of W.)
+Any finite product at common twist has ∅,∅ entry
+1 (loop cancellation, as in Section~sec:strip-physical).
Judicious imaginary-limit sampling reduces the simplicity on total spin
-zero to a base case $N=1$.
+zero to a base case N=1.
-Vector normalization. We set $w^xi=P^xi|∅>$,
-$ell^xi=<∅|P^xi$, both rational with empty entry $1$,
-and related by crossing $ell^xi(v)=w^{xi^{-1}}(-v)^T$. Exchange and
-fusion of $w$ follow as in Section~sec:strip-physical: at
-$v_q-v_pin{0,2L,b}$, the local $R_{pq}$ and its fusions act as stated.
+Vector normalization. We set w^xi=P^xi|∅>,
+ell^xi=<∅|P^xi, both rational with empty entry 1,
+and related by crossing ell^xi(v)=w^{xi^{-1}}(-v)^T. Exchange and
+fusion of w follow as in Section~sec:strip-physical: at
+v_q-v_pin{0,2L,b}, the local R_{pq} and its fusions act as stated.
-Degree bound. For each row $T_xi(t)$ on the vacuum line,
+Degree bound. For each row T_xi(t) on the vacuum line,
Fourier-Laurent expansion gives
[
T_xi(t)=I+sum_k[A_kcot(t+v_k+2L)+B'_kcot(t+v_k+3L)],
qquadsum_k(A_k+B'_k)=0.
]
-Applying the fusion equations to $T_j=T_xi(-v_p+jL)$ gives commuting-operator
-annihilator rows $mathcal A_calA_{p,0}=c_0-c_1+c_2+X_0c_2$ and
-$mathcal A_calA_{p,1}=c_1-c_2+c_3+X_1c_3$. Collecting into operator
-matrix $mathcal A$,calA$, one has
-$detmathcal A=(calA=(det H_{mathcal A}calA)P^xi$, where $H_{mathcal A}calA$
-is the cotangent kernel $r(u)=cot(u+3L)-cot(u+4L)+cot(u+5L)$.
-The scalar Pfaffian $J_N=Pf(H_{mathcal A}calA)prod_{j<k}p(v_k-v_j)$,
+Applying the fusion equations to T_j=T_xi(-v_p+jL) gives commuting-operator
+annihilator rows mathcal A_calA_{p,0}=c_0-c_1+c_2+X_0c_2 and
+mathcal A_calA_{p,1}=c_1-c_2+c_3+X_1c_3. Collecting into operator
+matrix mathcal A,calA, one has
+detmathcal A=(calA=(det H_{mathcal A}calA)P^xi, where H_{mathcal A}calA
+is the cotangent kernel r(u)=cot(u+3L)-cot(u+4L)+cot(u+5L).
+The scalar Pfaffian J_N=Pf(H_{mathcal A}calA)prod_{j<k}p(v_k-v_j),
p(u)=frac{sin(u+2L)sin(u-2L)cos u [sin(u+b)sin(u-b)]^2}
{sin^2u,sin(u+L)sin(u-L)},
-is a polynomial of trigonometric degree $le 3(N-1)$ per variable.
+is a polynomial of trigonometric degree le 3(N-1) per variable.
Next, the two-site computation
-J_2=cos u,[S t_*-(t_*^2/2)cos 2u],qquad t_*=-(1)/(1+S),
-
-identifies three simple roots modulo $pi$: one real at $uequivpi/2$,
-and two complex from $cos 2u=-1-sqrt2$. This gives generic simplicity
-at $v_j-v_kin{L,2L}$. Combining with
-fusion via $W$ and $Ω$, the total pole degree of $P^xi$ is
-bounded by $3(N-1)$, hence $d_N^i+d_N^{-i}le 3(N-1)$.
-
-Wheel zeros. Consider $Psi$ at $(v_1,v_2,v_3)=(u,u+2L,u+5L)$
-in the order $1,2,3$ with remaining list $Y$. Perturbing
-$v_2=u+2L+z$, $v_3=u+5L+z'$ and using~(eq:psi-fusion-2L):
-at $z=0$, $Psiinim(W_{12})$; on $z'=z$ and $z'=0$ we get
-specific columns $b_0,c_0$ proportional to $Ω_{23}$ and to
-$K_3 J^{rm spin}_1 Ω_{31}$ (composed with site-3 factors). The
-column is in $im(W_{12})$ only at specific spin coefficient
-identities; matching at spin $pm1$ then at spin $0$ forces both
-$gamma,alpha=0$ (for $xi=-i$), giving vanishing of order $ge2$.
-
-The Pfaffian normalization. The candidate $f_N$ of
+J_2=cos u,[S t_*-(t_*^2/2)cos 2u], t_*=-(1)/(1+S),
+
+identifies three simple roots modulo pi: one real at uequivpi/2,
+and two complex from cos 2u=-1-sqrt2. This gives generic simplicity
+at v_j-v_kin{L,2L}. Combining with
+fusion via W and Ω, the total pole degree of P^xi is
+bounded by 3(N-1), hence d_N^i+d_N^{-i}le 3(N-1).
+
+Wheel zeros. Consider Psi at (v_1,v_2,v_3)=(u,u+2L,u+5L)
+in the order 1,2,3 with remaining list Y. Perturbing
+v_2=u+2L+z, v_3=u+5L+z' and using~(eq:psi-fusion-2L):
+at z=0, Psiinim(W_{12}); on z'=z and z'=0 we get
+specific columns b_0,c_0 proportional to Ω_{23} and to
+K_3 J^{rm spin}_1 Ω_{31} (composed with site-3 factors). The
+column is in im(W_{12}) only at specific spin coefficient
+identities; matching at spin pm1 then at spin 0 forces both
+gamma,alpha=0 (for xi=-i), giving vanishing of order ge2.
+
+The Pfaffian normalization. The candidate f_N of
(eq:f_N) also satisfies the fusions~(eq:psi-fusion-2L),
-(eq:psi-fusion-b) (using $B(d)B(d-b)=-cos(d-L)cos(d-2L)$ and
-induction on $N$) and has the required degree, parity and symmetry.
-The base cases $f_0,f_1,f_2$ match by direct computation. A polynomial
-of degree $le N-1$ with fixed parity and $2(N-1)$ prescribed values
-modulo $pi$ is unique, giving $f_N=Psi_N^∅$.
+(eq:psi-fusion-b) (using B(d)B(d-b)=-cos(d-L)cos(d-2L) and
+induction on N) and has the required degree, parity and symmetry.
+The base cases f_0,f_1,f_2 match by direct computation. A polynomial
+of degree le N-1 with fixed parity and 2(N-1) prescribed values
+modulo pi is unique, giving f_N=Psi_N^∅.
-The remaining claims on $Psi_N$ (degree control, parity) follow from
+The remaining claims on Psi_N (degree control, parity) follow from
the fusion and normalized-vector structure.
-The equality of the exact degree to $3(N-1)$ is then obtained from
-divergence at $v_2-v_1in{b}$ in the double fusion~(eq:psi-fusion-b):
-the wheel factor there forces $d_N^ige N-1$ and $d_N^{-i}ge 2(N-1)$.
+The equality of the exact degree to 3(N-1) is then obtained from
+divergence at v_2-v_1in{b} in the double fusion~(eq:psi-fusion-b):
+the wheel factor there forces d_N^ige N-1 and d_N^{-i}ge 2(N-1).
Matching (eq:f_N) gives the exact scalar fusion identities
Psi_N(u,u+2L,U) &= c_1prod_{zin U}cos(z-u-L)cdot W_{12}Psi_{N-1}(u+L,U),
Psi_N(u,u+b,U) &= c_2prod_{zin U}cos(z-u-L)cos(z-u-2L)cdotΩ_{12}Psi_{N-2}(U),
-with $c_1=1$ (after normalization) and $c_2=2cos b$
-(cross-checked at $N=2,3$ and by induction).
+with c_1=1 (after normalization) and c_2=2cos b
+(cross-checked at N=2,3 and by induction).
The physical contraction.:
Proof. [Proof of Proposition~prop:physical-pairing]
Define
-$Z(v)=ell^{-i}(v)(prod_{jle m}K_j(delta))w^i(v)$,
-with $delta=i y$. Then $U=2f_N^2Z$ is a trigonometric polynomial
-of degrees $le 2(N-1)$ in each variable, $pi$-periodic by spin-sign
+Z(v)=ell^{-i}(v)(prod_{jle m}K_j(delta))w^i(v),
+with delta=i y. Then U=2f_N^2Z is a trigonometric polynomial
+of degrees le 2(N-1) in each variable, pi-periodic by spin-sign
covariance, and separately symmetric in the two groups (spin
conservation plus exchange).
-Both $U$ and $G_N$ (from~(eq:G-def)) share:
+Both U and G_N (from~(eq:G-def)) share:
+ separate symmetry in the two groups,
+ pi-periodicity in each v_j,
+ exact fusion to the smaller list at fused rapidities
+(v_{j+1}-v_j)in{0,pm2L,pm b} (with normalizations from f_N),
+ growth behavior at imaginary infinity by~(eq:psi-fusion-b),
+ initial conditions G_{N=0}=1, G_{N=1}=1 (identical to U).
-item separate symmetry in the two groups,
-item $pi$-periodicity in each $v_j$,
-item exact fusion to the smaller list at fused rapidities
-$(v_{j+1}-v_j)in{0,pm2L,pm b}$ (with normalizations from $f_N$),
-item growth behavior at imaginary infinity by~(eq:psi-fusion-b),
-item initial conditions $G_{N=0}=1$, $G_{N=1}=1$ (identical to $U$).
-
-By induction on $N$, $G_N=U$. Indeed, the difference $G_N-U$ has zero
+By induction on N, G_N=U. Indeed, the difference G_N-U has zero
degree after dividing by the fusion-denominator product and vanishes
-at each step. The base cases $N=0,1$ match.
-
-Evaluating at $v_1=...=v_N=0$ with the sandwiched $K_j(delta)$'s:
+at each step. The base cases N=0,1 match.
-item $mathcal Z_N$calZ_N$ at homogeneous $v$ has a natural expansion as an
+Evaluating at v_1=...=v_N=0 with the sandwiched K_j(delta)'s:
+ mathcal Z_NcalZ_N at homogeneous v has a natural expansion as an
oriented sum over polygon systems; the contractible and essential
-sectors get phases $i$ and $pm i$ (reversing conjugates), and the
-`left-side switch' factor $(-1)^{1_{Ain D'}}$ plus
-$delta^{1_{Ain D'}-1_{Bin D'}}$ from the sandwich. Reversing
-orientation inverts the factor, so a polygon contributes $0$ unless
-$A,B$ are separated, giving weight $y+y^{-1}$ exactly.
-item The tile weights in $G_N$ without phases are honeycomb path
+sectors get phases i and pm i (reversing conjugates), and the
+`left-side switch' factor (-1)^{1_{Ain D'}} plus
+delta^{1_{Ain D'}-1_{Bin D'}} from the sandwich. Reversing
+orientation inverts the factor, so a polygon contributes 0 unless
+A,B are separated, giving weight y+y^{-1} exactly.
+ The tile weights in G_N without phases are honeycomb path
weights.
-item The empty system gives $1$, matching $(∅,∅)$
+ The empty system gives 1, matching (∅,∅)
entry.
-For positive $y$, exhaustion proves $mathcal Z_N=G_N/(2f_N^2)|_calZ_N=G_N/(2f_N^2)|_{v=0}$,
+For positive y, exhaustion proves mathcal Z_N=G_N/(2f_N^2)|_calZ_N=G_N/(2f_N^2)|_{v=0},
and the equality extends to a Laurent polynomial identity.
@@ -1630,7 +1622,7 @@ Section~c6:periodic-polynomial expressed the physical cylinder
partition as a finite trigonometric ratio. We now expand that ratio as
a grand-canonical sum over particles, and control the sum by a
coercivity bound on the kernel. The output is a rigorous upper bound
-on $log|P_m|$ (Proposition~prop:gas-upper) which
+on log|P_m| (Proposition~prop:gas-upper) which
Section~c6:subheading-09 saturates.
Normalization and the target.:
@@ -1639,13 +1631,13 @@ Put
P_m(e^r)=frac{G(e^r)}{C^mD(0)^{m(m-1)}}.
-With $L=pi/8$, $b=3L$, $C=2-sqrt2$, $widehatrho(p)=1/(2cosh Lp-1)$,
-and $V(s)$ the real log of $R_s(is)R_c(is)$ for
-$R_s(z)=-1/[sin(z-L)sin(z+L)]$, $R_c(z)=cos^2z$, put
-$I=int_{mathbb R}bbRV(s)rho(s),ds$.
+With L=pi/8, b=3L, C=2-sqrt2, widehatrho(p)=1/(2cosh Lp-1),
+and V(s) the real log of R_s(is)R_c(is) for
+R_s(z)=-1/[sin(z-L)sin(z+L)], R_c(z)=cos^2z, put
+I=int_{mathbb R}bbRV(s)rho(s),ds.
[Homogeneous gas upper bound]
-As $mto∞$, locally uniformly in $|Im r|<.54pi$:
+As mto∞, locally uniformly in |Im r|<.54pi:
log|P_m(e^r)|le m^2I+(frac38+(2)/(3pi^2)Re r^2)log m+o(log m).
@@ -1654,35 +1646,35 @@ log|P_m(e^r)|le m^2I+(frac38+(2)/(3pi^2)Re r^2)log m+o(log m).
Residue particles.:
Each lowering of a site label in the finite double-pole formula
-produces one or two residue particles. A particle $d$ carries a color
-sign $q_din{pm1}$, a species, a contour, and a real position $s_d$.
+produces one or two residue particles. A particle d carries a color
+sign q_din{pm1}, a species, a contour, and a real position s_d.
-The elementary mode potential is $e^{mV(w)}$, and the pair weights are
+The elementary mode potential is e^{mV(w)}, and the pair weights are
t_{gh}(w)=frac{R_{gh}(iw)}{R_{gh}(iw+L)R_{gh}(iw-L)},
-with measure $tau dz/(2pi i)$ and $tau=2sin(2L)/sqrt C$.
+with measure tau dz/(2pi i) and tau=2sin(2L)/sqrt C.
-After periodization (replacing each $sinh(w+ijL)$ by
-$frac12S_T(w+ijL)$ with $S_T(w)=2sinh wprod_{nge1}(1-e^{-2nT+2w})
-(1-e^{-2nT-2w})$, and multiplying the integrand by
-$sum_le^{-2Tl^2+4lsum q_jw_j}$) we obtain a $T$-periodic
-$P_{m,T}(e^r)$, agreeing with $P_m(e^r)$ at $e^{-2T}=0$. We work on
-$P_{m,T}$ and return by maximum modulus in the nome $e^{-T}$.
+After periodization (replacing each sinh(w+ijL) by
+frac12S_T(w+ijL) with S_T(w)=2sinh wprod_{nge1}(1-e^{-2nT+2w})
+(1-e^{-2nT-2w}), and multiplying the integrand by
+sum_le^{-2Tl^2+4lsum q_jw_j}) we obtain a T-periodic
+P_{m,T}(e^r), agreeing with P_m(e^r) at e^{-2T}=0. We work on
+P_{m,T} and return by maximum modulus in the nome e^{-T}.
Species and offsets.:
After contour manipulation and fusion, five species remain in each
-color. Their weights $w_i$ (exponent of $y$), offsets $delta_i$
-(multiples of $L$), and chemical activities $sigma_i$ are:
+color. Their weights w_i (exponent of y), offsets delta_i
+(multiples of L), and chemical activities sigma_i are:
{c|ccccc}
-$i$ & $0$ & $1$ & $2$ & $3$ & $4$
-hline
-label & $S$ & $D$ & $I_+'$ & $I_-'$ & $F$
-$w_i$ & $1$ & $1$ & $-1$ & $1$ & $2$
-$12delta_i$ & $6$ & $1$ & $5$ & $-3$ & $-1$
-$sigma_i$ & $0$ & $1$ & $1$ & $-1$ & $0$
+i & 0 & 1 & 2 & 3 & 4
+
+label & S & D & I_+' & I_-' & F
+w_i & 1 & 1 & -1 & 1 & 2
+12delta_i & 6 & 1 & 5 & -3 & -1
+sigma_i & 0 & 1 & 1 & -1 & 0
The neutral-winding constraint on total particle counts is
@@ -1692,91 +1684,90 @@ sum_{gin{c,-}}sum_{i=1}^5w_inu_{g,i}=0.
The interaction kernel.:
-The particle kernel for two particles at real frequency $p$ is
+The particle kernel for two particles at real frequency p is
Z^{(c)}_{ij}(p)=(2pi)/(p)e^{(delta_i-delta_j)Lp}F^{(c)}_{ij}(e^{Lp}),
-with $F^{(c)}_{ij}(z)$ expressed through an auxiliary function $H(a;sigma)$
+with F^{(c)}_{ij}(z) expressed through an auxiliary function H(a;sigma)
that sums residues on the unit circle:
H(a;sigma)(z)=sum_{j=0}^7h_jfrac{z^{[a+j]_8}}{z^8-1}+(1-sigma)/(2)h_{[-a]_8},
-quad h=(2,-1,1,-2c,2c,-2c,1,-1).
+ h=(2,-1,1,-2c,2c,-2c,1,-1).
-Let $H_*=H(0;0)$, $A_0=sum_{ain(2)}H(a;1)$, $A_1=sum_{ain(2,3)}H(a;-1)$,
-$A_2=1$, $A_3=-1$, $A_4=A_1+A_3$. For $0le i,j<4$, set
+Let H_*=H(0;0), A_0=sum_{ain(2)}H(a;1), A_1=sum_{ain(2,3)}H(a;-1),
+A_2=1, A_3=-1, A_4=A_1+A_3. For 0le i,j<4, set
F^{(c)}_{ij}(z)=P_{ij}(z)+frac{A_i(z)A_j(z^{-1})}{H_*(z)}
+(-1+2c)sgn(i-j){mathbf 1}bf1_{{i,j}={1,3}},
-$F_{i4}=F_{i1}+F_{i3}+{mathbf 1}bf1_{i=3}-{mathbf 1}bf1_{i=1}$,
-$F_{4i}(z)=-F_{i4}(z^{-1})$, $F_{44}=F_{11}+F_{13}+F_{31}+F_{33}$,
-where $P_{ij}$ collects lower-derivative residue rules
-$P_{ij}(z)=sum_{ain b_i,kin b_j}H(a-k;sgn(j-i))$ for $i,j<2$
-and $sgn(j-i)$ for $i,jge2$, and $0$ otherwise.
+F_{i4}=F_{i1}+F_{i3}+{mathbf 1}bf1_{i=3}-{mathbf 1}bf1_{i=1},
+F_{4i}(z)=-F_{i4}(z^{-1}), F_{44}=F_{11}+F_{13}+F_{31}+F_{33},
+where P_{ij} collects lower-derivative residue rules
+P_{ij}(z)=sum_{ain b_i,kin b_j}H(a-k;sgn(j-i)) for i,j<2
+and sgn(j-i) for i,jge2, and 0 otherwise.
-On the two $R$-lines (before fusion), the matrix symbol has a clean form.
-Let $K=K_{RR}$ and $widehatrho(p)=1/(2cosh Lp-1)$; then
+On the two R-lines (before fusion), the matrix symbol has a clean form.
+Let K=K_{RR} and widehatrho(p)=1/(2cosh Lp-1); then
-K=(1-e(p))K_0,quad
+K=(1-e(p))K_0,
e(p)=(sinh(3Lp)-sinh(2Lp))/(sinh(4Lp))in(0,1),
K_0=(2pi)/(p)coth(bp)&-csch(bp)
--csch(bp)&coth(bp),quad V_R=K{mathbf 1}bf1widehatrho.
+-csch(bp)&coth(bp), V_R=K{mathbf 1}bf1widehatrho.
-(Here $V_R$ denotes the potential contribution.)
+(Here V_R denotes the potential contribution.)
Coercivity.:
-The key analytic input is a positive-definite bound on $Z$.
+The key analytic input is a positive-definite bound on Z.
[Coercivity]
-Put $h_*=3pi b/4$ and $alpha=2pi^2/h_*$, and let $W=(w,w)^t$ be the
-weight column. Then for every real $pne0$ there is $eta>0$ with
+Put h_*=3pi b/4 and alpha=2pi^2/h_*, and let W=(w,w)^t be the
+weight column. Then for every real pne0 there is eta>0 with
Z_{rm s}(p)succcurlyeq(etaalpha)/(p^2)WW^t+ε_0(1+p^2)^{-1}{bf I},
-for some $ε_0>0$ independent of $T$ and $m$, where $Z_s=(Z+Z^t)/2$.
+for some ε_0>0 independent of T and m, where Z_s=(Z+Z^t)/2.
Proof. [Sketch]
-On each $R$-line, $K=(1-e(p))K_0$ with $e(p)in(0,1)$ and $K_0$
-positive definite for $pne0$. The leading behavior at small $p$ is
-$K_0sim2pi/(p)(coth bp{bf I}-cschbp,P)$ where
-$P$ is the color-swap permutation matrix; the eigen-decomposition is
-common and gives the $WW^t/p^2$ leading term. Away from $p=0$, the
-residual is bounded below by $ε_0(1+p^2)^{-1}{bf I}$ after
-partial-fraction expansion of the explicit $F$-rules.
+On each R-line, K=(1-e(p))K_0 with e(p)in(0,1) and K_0
+positive definite for pne0. The leading behavior at small p is
+K_0sim2pi/(p)(coth bp{bf I}-cschbp,P) where
+P is the color-swap permutation matrix; the eigen-decomposition is
+common and gives the WW^t/p^2 leading term. Away from p=0, the
+residual is bounded below by ε_0(1+p^2)^{-1}{bf I} after
+partial-fraction expansion of the explicit F-rules.
-Z(p)=alpha WW^t/p^2+M_*/p+O(1)quad(pto0),
-quad M_* real skew.
+Z(p)=alpha WW^t/p^2+M_*/p+O(1)(pto0),
+ M_* real skew.
-Z(p)=(2pi)/(|p|){mathbf I}bfI+2ff^t+O(e^{-ε|p|})quad(|p|to∞),quad f={mathbf e}bfe_4.
+Z(p)=(2pi)/(|p|){mathbf I}bfI+2ff^t+O(e^{-ε|p|})(|p|to∞), f={mathbf e}bfe_4.
The grand sum.:
[Grand sum]
-For every integer $mge1$ and every sufficiently large real $T$,
-$P_{m,T}(e^r)$ equals
+For every integer mge1 and every sufficiently large real T,
+P_{m,T}(e^r) equals
sum_{{nu_{g,i}:,(eq:winding-neutrality)}}
frac{1}{prod_{g,i}nu_{g,i}!}
,e^{ccdot 2pi iTr,sigma^tnu/(4h_*)-S_{rm bulk}(T)-S_{rm gas}[nu;r]},
with:
-
-item $S_{rm bulk}(T)=(1)/(2T)sum_{pne0}widehatwidehat J^tK(p)J^tK(p)widehatwidehat JJ,widehatrho(p)^2
--(1)/(2)(T/3-2log2)(M_A-M_B)^2$,
+ S_{rm bulk}(T)=(1)/(2T)sum_{pne0}widehatwidehat J^tK(p)J^tK(p)widehatwidehat JJ,widehatrho(p)^2
+-(1)/(2)(T/3-2log2)(M_A-M_B)^2,
a real quadratic form;
-item $S_{rm gas}[nu;r]=-(1)/(2T)sum_{pne0}sum_{d,e}Z_{g_de_d}(p)e^{-ip(s_d-s_e)}-(1)/(T)sum_{linear}...$,
+ S_{rm gas}[nu;r]=-(1)/(2T)sum_{pne0}sum_{d,e}Z_{g_de_d}(p)e^{-ip(s_d-s_e)}-(1)/(T)sum_{linear}...,
the pairwise and single-particle interactions against real positions
-$s_din(-T/2,T/2)$;
-item $ε_nuin{pm1}$ is a fixed phase depending only on the multiplicity pattern;
-item $c$ is a fixed scalar.
+s_din(-T/2,T/2);
+ ε_nuin{pm1} is a fixed phase depending only on the multiplicity pattern;
+ c is a fixed scalar.
The sum and integral converge absolutely, and the sum localizes on
@@ -1787,13 +1778,13 @@ max_d|Re s_d|lesssimlog m+const.
Convergence and localization.:
Proof. [Sketch]
-Proposition~prop:coercivity bounds the off-diagonal $Z_s$ terms
+Proposition~prop:coercivity bounds the off-diagonal Z_s terms
from below by a positive quadratic form. After regularizing the diagonal,
-the sum over $nu_{g,i}$ is bounded by a geometric series in
-$sum_{g,i}nu_{g,i}$ weighted by a Gaussian-decaying kernel in the
-positions $s_d$ and a bounded activity from the chemical potential. For any configuration with a particle at $|Re s_d|>log m+const$,
-the one-particle suppression from $S_{rm bulk}$ dominates the local
-gain, and the sum over such configurations is at most $e^{-ε m}$.
+the sum over nu_{g,i} is bounded by a geometric series in
+sum_{g,i}nu_{g,i} weighted by a Gaussian-decaying kernel in the
+positions s_d and a bounded activity from the chemical potential. For any configuration with a particle at |Re s_d|>log m+const,
+the one-particle suppression from S_{rm bulk} dominates the local
+gain, and the sum over such configurations is at most e^{-ε m}.
Consequently the whole sum is dominated by the localized region
(eq:gas-localization).
@@ -1802,16 +1793,16 @@ Consequently the whole sum is dominated by the localized region
Saturation at the two physical parameters.:
-We have the upper bound $log|P_m(e^r)|le m^2I+(3/8+2Re(r^2)/(3pi^2))
-log m+o(log m)$ for $|Im r|<.54pi$ (Proposition~prop:gas-upper),
-plus an exact evaluation of $P_m(i)$ via a determinant at the "empty
+We have the upper bound log|P_m(e^r)|le m^2I+(3/8+2Re(r^2)/(3pi^2))
+log m+o(log m) for |Im r|<.54pi (Proposition~prop:gas-upper),
+plus an exact evaluation of P_m(i) via a determinant at the "empty
polygon" value. We now show both bounds are sharp and take their ratio.
The saturation statement.:
-[Saturation at $y=1$ and $y=i$]
+[Saturation at y=1 and y=i]
-log P_m(1)=Im^2+frac38log m+o(log m),qquad
+log P_m(1)=Im^2+frac38log m+o(log m),
log P_m(i)=Im^2+frac5{24}log m+o(log m).
Equivalently, the nesting ratio satisfies
@@ -1820,45 +1811,45 @@ mathcal Z_calZ_{m,m}(1)=(P_m(1))/(P_m(i))=m^{1/6+o(1)}.
-The reference determinant at $y=i$.:
+The reference determinant at y=i.:
-At $y=i$ the finite polynomial $G$ becomes $2f_N^2$: only the empty
+At y=i the finite polynomial G becomes 2f_N^2: only the empty
polygon system contributes. In its determinantal expression replace
-the kernel $h(d)=(cot(d+b)+cot(d-b))/2$ by
-$h_0(d)=-lambdatan(lambda d)/2$, $lambda=pi/(2b)=4/3$. The
+the kernel h(d)=(cot(d+b)+cot(d-b))/2 by
+h_0(d)=-lambdatan(lambda d)/2, lambda=pi/(2b)=4/3. The
normalized homogeneous value of this modified partition evaluates
exactly, by Schur's Pfaffian identity, to
P_0^{(lambda)}=big[lambda^4sin^8(b)/D(0)big]^{m^2}=e^{Im^2}.
-Indeed, Schur's Pfaffian of $G_{jk}=(X_j-X_k)/(X_j+X_k)$ with
-$X_j=e^{2ilambda v_j}$ equals $prod_{i<j}(X_i-X_j)^2/
-prod(X_i+X_j)^2$ (clearing denominators, both sides alternate with
+Indeed, Schur's Pfaffian of G_{jk}=(X_j-X_k)/(X_j+X_k) with
+X_j=e^{2ilambda v_j} equals prod_{i<j}(X_i-X_j)^2/
+prod(X_i+X_j)^2 (clearing denominators, both sides alternate with
the same Vandermonde degree; the coefficient is fixed by a base case),
-and $B(u)tan(lambda u)to-lambdasin^2b$ at coincidence while
-$D(0)=Csin^4b$.
+and B(u)tan(lambda u)to-lambdasin^2b at coincidence while
+D(0)=Csin^4b.
-We are reduced to bounding the ratio $P_m(i)/P_0^{(lambda)}$.
+We are reduced to bounding the ratio P_m(i)/P_0^{(lambda)}.
-Bounding the ratio at $y=i$.:
+Bounding the ratio at y=i.:
-Write $P_m(i)/P_0^{(lambda)}$ as the product over $m$ "duplicate
-pairs" $v_p,v_p+L$ of the linear-perturbation ratios
+Write P_m(i)/P_0^{(lambda)} as the product over m "duplicate
+pairs" v_p,v_p+L of the linear-perturbation ratios
prod_{k=1}^m(det(1+Delta_k))/(det(1)),
-where $Delta_k$ is the rank-one modification between the $h$ and $h_0$
-kernels restricted to the $k$-th pair.
+where Delta_k is the rank-one modification between the h and h_0
+kernels restricted to the k-th pair.
Weyl chamber decomposition. Decompose the integration domain by
-the partition of ${1,...,m}$ into the subsets $Ω^A$, where $A$
-is the unique subset containing the up-to-relabeling largest $|z_j|$.
-On each $Ω^A$, moving the rapidities with $jin A$ to imaginary
-infinity is pole-free within the chamber, and the two kernels (for $h$
-and $h_0$) differ by a bounded factor with a Sobolev/$L_p$ estimate
+the partition of {1,...,m} into the subsets Ω^A, where A
+is the unique subset containing the up-to-relabeling largest |z_j|.
+On each Ω^A, moving the rapidities with jin A to imaginary
+infinity is pole-free within the chamber, and the two kernels (for h
+and h_0) differ by a bounded factor with a Sobolev/L_p estimate
from the linear-differential operator computation.
-For each $k$, one gets uniformly
+For each k, one gets uniformly
big|det(1+Delta_k)big|le e^{-ε,dist(A_k,
∂chamber)}.
@@ -1870,28 +1861,28 @@ logbig|P_m(i)/P_0^{(lambda)}big|lefrac5{24}log m+O(1).
Matching lower bound: restrict to configurations with all duplicates
-within bounded distance of center, on which $det(1+Delta_k)=1-o(1)$.
-This gives $|P_m(i)/P_0^{(lambda)}|ge m^{5/24-o(1)}$, hence
+within bounded distance of center, on which det(1+Delta_k)=1-o(1).
+This gives |P_m(i)/P_0^{(lambda)}|ge m^{5/24-o(1)}, hence
log P_m(i)=Im^2+frac5{24}log m+o(log m).
-Propagation to $y=1$.:
+Propagation to y=1.:
-Set $r=log y$ and put
+Set r=log y and put
f_m(r)=(log|P_m(e^r)|-Im^2)/(log m)
-(frac38+(2Re(r^2))/(3pi^2)).
-From Proposition~prop:gas-upper, $f_mle o(1)$ locally
-uniformly in $|Im r|<.54pi$. From (eq:P-m-i-final) at
-$r=ipi/2$ (i.e. $y=i$), $f_m(ipi/2)to0$.
+From Proposition~prop:gas-upper, f_mle o(1) locally
+uniformly in |Im r|<.54pi. From (eq:P-m-i-final) at
+r=ipi/2 (i.e. y=i), f_m(ipi/2)to0.
-The function $f_m$ is subharmonic. Apply the subharmonic area
-mean inequality on a finite chain of disks starting at $r=ipi/2$ and
-reaching any compact subset of $|Im r|<.54pi$: the two ends of the
-chain give $f_mto0$ in local $L^1$ throughout the strip. In
-particular at $r=0$:
+The function f_m is subharmonic. Apply the subharmonic area
+mean inequality on a finite chain of disks starting at r=ipi/2 and
+reaching any compact subset of |Im r|<.54pi: the two ends of the
+chain give f_mto0 in local L^1 throughout the strip. In
+particular at r=0:
log P_m(1)=Im^2+frac38log m+o(log m).
@@ -1908,109 +1899,108 @@ as claimed.
Crossings, cylinder barriers, and positive nesting.:
Section~c6:strip-asymptotics gave the strip-mass estimate
-$B_kasymp k^{-1/4}$. Here we extract (a) escape and confinement
+B_kasymp k^{-1/4}. Here we extract (a) escape and confinement
estimates, (b) essential-polygon formation on a cylinder, (c) positive
polygon nesting, and (d) a boundary-occupancy identity. These are the
geometric inputs for Section~sec:sewing.
-All constants are uniform in $N$, $h$, and $H$.
+All constants are uniform in N, h, and H.
The critical weight at the port convention.:
Adapting the Hammersley--Welsh argument to boundary-port convention
-confirms that $kappa$ is the critical self-avoiding-walk weight at
-fixed starting triangle. The route uses strip bridges: for $x<kappa$
-the mass of weak bridges of height $k$ is at most $Ce^{-ck}$, and
+confirms that kappa is the critical self-avoiding-walk weight at
+fixed starting triangle. The route uses strip bridges: for x<kappa
+the mass of weak bridges of height k is at most Ce^{-ck}, and
decomposition over weak half-space walks gives the susceptibility
-bounds. Constants depend only on $x$.
+bounds. Constants depend only on x.
Boundary crossings.:
[Boundary crossings]
-Fix a convex pure-sided tiled domain $D$ and a port $p$ on its
+Fix a convex pure-sided tiled domain D and a port p on its
boundary (including strips and half-planes by exhaustion). Then:
-
-item The critical mass of completed paths of diameter $ge r$ is at
-most $C(1+r)^{-1/4}$.
-item The half-plane mass $K_l$ from a base port to another at lattice
-distance $l$ is decreasing in $l$ and satisfies $K_lasymp l^{-5/4}$.
+ The critical mass of completed paths of diameter ge r is at
+most C(1+r)^{-1/4}.
+ The half-plane mass K_l from a base port to another at lattice
+distance l is decreasing in l and satisfies K_lasymp l^{-5/4}.
Proof. [Sketch]
Contour comparison. The signed exit identity (Section~c6:strip-flux
-with $beta=3/8$) gives
+with beta=3/8) gives
sum_{gamma:pto∂ D}kappa^{ell(gamma)}
-cos(beta W(gamma))=1,quad clecos(beta W(gamma))le1.
+cos(beta W(gamma))=1, clecos(beta W(gamma))le1.
-Restricting to a convex subdomain and using $c=cos(betapi)>0$ gives
+Restricting to a convex subdomain and using c=cos(betapi)>0 gives
the first assertion.
-Triangle escape. For a centered triangle $T(p,n_j,t)$ of width
-$t$, the exit mass $S(t)$ to each sloping side is comparable to
-$(1+t)^{-1/4}$. The upper bound comes from the strip embedding; the
+Triangle escape. For a centered triangle T(p,n_j,t) of width
+t, the exit mass S(t) to each sloping side is comparable to
+(1+t)^{-1/4}. The upper bound comes from the strip embedding; the
lower bound from the base-return identity
-$2cos(betapi/3)S=1-cA_T$ and the diameter bound.
+2cos(betapi/3)S=1-cA_T and the diameter bound.
-Monotonicity. A non-crossing identity for $K$ and an intermediate
-port between sources $a$ and $Q=P+1$ gives
+Monotonicity. A non-crossing identity for K and an intermediate
+port between sources a and Q=P+1 gives
(K_{P-a}-K_{Q-a})sin(betapi)=D_1sin(2betapi)ge0,
using the flux rule with an artificial arc below the base line. Combined
-with the diameter bound over a window of length $asymp l$: $K_llesssim l^{-5/4}$.
+with the diameter bound over a window of length asymp l: K_llesssim l^{-5/4}.
-Deep exits. For an exit from $T(a,n_0,t)$ through the side with
-outward normal $n_1$, at depth $q=n_0cdot(p_1-a)$,
+Deep exits. For an exit from T(a,n_0,t) through the side with
+outward normal n_1, at depth q=n_0cdot(p_1-a),
mathop{average}_{tin[alpha R,AR]}sum_{qgedelta R}k(t,q)
ge c_{alpha,A,delta}R^{-1/4},
-for $delta$ small, by translating along the base to align the exit
+for delta small, by translating along the base to align the exit
ports and cutting strictly before all starts.
Three exits and matching lower bound. Iterating the deep-exit
-construction at $p_1,p_2$ builds a three-exit path. Averaging over $t$
-and $R$ gives mass $gtrsim R^{-3/4}$ of paths with $qgedelta R$ and
-$Xgeeta R$, whence a second-moment bound
+construction at p_1,p_2 builds a three-exit path. Averaging over t
+and R gives mass gtrsim R^{-3/4} of paths with qgedelta R and
+Xgeeta R, whence a second-moment bound
mu[Y^2]le CR^{-5/4},
-using the difference bridges with $B_{|t-t'|/d_0}$. Cauchy--Schwarz gives
-$mu[Y>0]gtrsim R^{-1/4}$ and the matching lower bound for $K_l$ by
+using the difference bridges with B_{|t-t'|/d_0}. Cauchy--Schwarz gives
+mu[Y>0]gtrsim R^{-1/4} and the matching lower bound for K_l by
monotonicity.
Narrow-channel bridges.:
[Narrow-channel bridges]
-For every sufficiently large absolute constant $A$, there are
-constants $C,R_0>0$ such that, for every strip of height $H$, every
-starting port on one cap, and every integer $Mge1$ with $H/Mge R_0$,
+For every sufficiently large absolute constant A, there are
+constants C,R_0>0 such that, for every strip of height H, every
+starting port on one cap, and every integer Mge1 with H/Mge R_0,
the critical mass of bridges whose transverse displacement from the
-starting port stays at most $4AH/M$ throughout is at least
+starting port stays at most 4AH/M throughout is at least
e^{-CM}(H/M)^{-1/4}.
Proof. [Sketch]
-Put $s=H/M$ and choose $M-1$ internal levels independently in windows
-of length $asymp s$ around equally spaced cuts. Each piece has height
-between $s/2$ and $2s$ and transverse excursion $le As$, with
+Put s=H/M and choose M-1 internal levels independently in windows
+of length asymp s around equally spaced cuts. Each piece has height
+between s/2 and 2s and transverse excursion le As, with
endpoint shift directed towards the original axis. Reflection retains
at least half the mass after the excursion restriction; the strip
-bound on each piece gives $cs^{-1/4}$; Cauchy--Schwarz over the lists
+bound on each piece gives cs^{-1/4}; Cauchy--Schwarz over the lists
of internal levels gives (eq:channel-bridge-mass).
Essential polygons in a slab.:
[Slab essential polygons]
-There are absolute constants $c,C>0$ such that, for every sufficiently
-large integer $N$ and every horizontal slab $S$ of height $CN$ on the
-cylinder of period $N$:
+There are absolute constants c,C>0 such that, for every sufficiently
+large integer N and every horizontal slab S of height CN on the
+cylinder of period N:
sum_{gammasubset S, gamma essential simple}
kappa^{ell(gamma)}ge c.
@@ -2019,10 +2009,10 @@ kappa^{ell(gamma)}ge c.
Proof. [Sketch]
Build confined arches via collar decompositions at two ports
-$p,q$ separated by $Din[N/2-ε N,N/2+ε N]$. Each collar
+p,q separated by Din[N/2-ε N,N/2+ε N]. Each collar
uses (eq:deep-exit), (eq:channel-bridge-mass), and one
-$S(t)$ factor, so a collar pair has mass $gtrsim N^{-3/2}$, and the
-connector adds another $N^{-5/4}$ factor by the $K_l$ estimate.
+S(t) factor, so a collar pair has mass gtrsim N^{-3/2}, and the
+connector adds another N^{-5/4} factor by the K_l estimate.
Averaging over the collar-cut lists and Cauchy--Schwarz gives a
slab-mass bound independent of the period. Joining upper and lower
confined arches and averaging over their shared base level gives
@@ -2031,17 +2021,17 @@ confined arches and averaging over their shared base level gives
Positivity of the negative polymer polynomial.:
-Let $D$ be the finite cylinder between horizontal row-boundary caps,
-with height $H=h d_0$ and quotient by $N$ steps. Let $mathcal P$calP$ be
-its set of simple polygons. For $0lebeta'le3/8$, put
-$alpha=pibeta'/3$, $x=(2cosalpha)^{-1}$, and
+Let D be the finite cylinder between horizontal row-boundary caps,
+with height H=h d_0 and quotient by N steps. Let mathcal PcalP be
+its set of simple polygons. For 0lebeta'le3/8, put
+alpha=pibeta'/3, x=(2cosalpha)^{-1}, and
omega(L)=x^{|L|}cdot
-2cos(4alpha),&L contractible,
-2cos(2alpha),&L essential.
-For $mathcal A⊆calA⊆mathcal P$:calP$:
+For mathcal A⊆calA⊆mathcal P:calP:
Q(mathcal A)=calA)=sum_{substack{mathcal F⊆calF⊆mathcal AcalA
polygons in mathcal FcalF disjoint}}
@@ -2049,13 +2039,13 @@ prod_{Linmathcal F}calFomega(L).
[Positivity]
-For every $Nge1$, $hge1$, $0lebeta'le3/8$, $mathcal A⊆calA⊆mathcal P$:calP$:
-$Q(mathcal A)>0$.calA)>0$.
+For every Nge1, hge1, 0lebeta'le3/8, mathcal A⊆calA⊆mathcal P:calP:
+Q(mathcal A)>0.calA)>0.
Proof. [Sketch]
-Short periods $N=1$ or $h=1$ follow from the explicit row-cycle
-formulas. For $N,hge2$: elementary polygons (six-centre loops around
+Short periods N=1 or h=1 follow from the explicit row-cycle
+formulas. For N,hge2: elementary polygons (six-centre loops around
interior lattice points) form a connected overlap family meeting every
triangle centre, so the incompatibility graph is connected. The signed
exit identity gives
@@ -2063,16 +2053,16 @@ exit identity gives
Q(mathcal P)=calP)=sum_{gamma:pto∂ D}x^{|gamma|}
e^{ibeta'W(gamma)}Q({L:Lcapgamma=∅}).
-Continue from $beta'=0$, $x=1/2$ where defect terms are nonnegative,
-to $beta'=3/8$. A minimal-zero subset $mathcal A$calA$ must be all of
-$mathcal P$calP$ by connectedness; but a fixed bridge through every level
+Continue from beta'=0, x=1/2 where defect terms are nonnegative,
+to beta'=3/8. A minimal-zero subset mathcal AcalA must be all of
+mathcal PcalP by connectedness; but a fixed bridge through every level
gives a positive residual polynomial, contradiction.
Deletion and free-energy bounds.:
-At the endpoint parameters replace $omega(L)$ by the essential-only
-weight $-tkappa^{|L|}$ with $tin[0,sqrt2]$. Positivity holds by
+At the endpoint parameters replace omega(L) by the essential-only
+weight -tkappa^{|L|} with tin[0,sqrt2]. Positivity holds by
multiaffinity. The deletion ratio is
H_H(gamma;t)geexpBig(t!sum_{ell essential, ellcapgammane∅}!kappa^{|ell|}Big).
@@ -2091,33 +2081,32 @@ H_H(ell;t),dtle CH/N.
Cylinder barriers and height tails.:
[Barriers and tails]
-On the infinite cylinder of period $N$:
-
-item The individual critical weights of essential polygons meeting a
+On the infinite cylinder of period N:
+ The individual critical weights of essential polygons meeting a
fixed horizontal level have bounded sum.
-item From a fixed bottom port in a semi-infinite cylinder, arches
-reaching height $s$ have mass at most $Ce^{-cs/N}$.
-item For $sge1$, all polygons meeting a fixed level with height span
-in $[s,2s]$ have total mass at most $poly(N,s)
-e^{-cs/N}$.
+ From a fixed bottom port in a semi-infinite cylinder, arches
+reaching height s have mass at most Ce^{-cs/N}.
+ For sge1, all polygons meeting a fixed level with height span
+in [s,2s] have total mass at most poly(N,s)
+e^{-cs/N}.
Proof. [Sketch]
Combine (eq:cylinder-deletion-ratio) with
-(eq:slab-essential-mass). A polygon of span $s$ meets order
-$s/(CN)$ disjoint slabs, each forcing positive-size cost:
-$H_H(gamma;t)gtrsim e^{c's/N}$. Sum by spans between $(jN,(j+1)N)$.
-The case $N=2$ uses an explicit two-strip surgery on the two choices of
-upward-centre; $N=1$ is a row-cycle count.
+(eq:slab-essential-mass). A polygon of span s meets order
+s/(CN) disjoint slabs, each forcing positive-size cost:
+H_H(gamma;t)gtrsim e^{c's/N}. Sum by spans between (jN,(j+1)N).
+The case N=2 uses an explicit two-strip surgery on the two choices of
+upward-centre; N=1 is a row-cycle count.
Local nesting and boundary occupancy.:
[Local nesting]
-Let $Z(R)$ be the positive partition sum of pairwise disjoint simple
-polygons in the plane, factor $2$, critical weight, supported within
-radius $R$ of a fixed triangular-lattice point, including the empty
+Let Z(R) be the positive partition sum of pairwise disjoint simple
+polygons in the plane, factor 2, critical weight, supported within
+radius R of a fixed triangular-lattice point, including the empty
system. Then
Z(R)=R^{1/12+o(1)}.
@@ -2130,15 +2119,15 @@ cost by (eq:cylinder-deletion-ratio). Contractible separators
lift to plane polygons around the marked point; the barrier bounds
(eq:cylinder-barriers) discard those with excessive vertical or
horizontal span. Systems inject. Hence
-$mathcal ZcalZlesssim Z(CN^{1+2xi})^2$ for any $xi>0$, and combining
-with (eq:nesting-ratio) and taking $xi↓0$ gives
+mathcal ZcalZlesssim Z(CN^{1+2xi})^2 for any xi>0, and combining
+with (eq:nesting-ratio) and taking xi↓0 gives
(eq:Z-R).
[Boundary occupancy]
-Let $D$ be a finite convex tiled domain and let $e$ be an internal
-honeycomb link crossing a tiling edge $[y,z]$. If $U_D(e)$ is the
-critical mass of completed external paths using $e$ with ordered free
+Let D be a finite convex tiled domain and let e be an internal
+honeycomb link crossing a tiling edge [y,z]. If U_D(e) is the
+critical mass of completed external paths using e with ordered free
boundary ports, then
U_D(e)asymp Z_D(y)+Z_D(z)
@@ -2148,10 +2137,10 @@ with absolute constants.
Proof. [Sketch]
The cut-link identity gives a signed exit mass
-$S_e=Z_e+(M_y+M_z)/sqrt2$, where $Z_e$ is the enclosing-polygon
-partition in the cut domain and $M_y,M_z$ are contributions of a
-polygon that uses $e$ with $y$ or $z$ inside ($Z_D(y)=Z_e+2M_y$,
-$Z_D(z)=Z_e+2M_z$). Reversing path prefixes and summing over sources
+S_e=Z_e+(M_y+M_z)/sqrt2, where Z_e is the enclosing-polygon
+partition in the cut domain and M_y,M_z are contributions of a
+polygon that uses e with y or z inside (Z_D(y)=Z_e+2M_y,
+Z_D(z)=Z_e+2M_z). Reversing path prefixes and summing over sources
yields (eq:occupancy).
@@ -2167,7 +2156,7 @@ in the sewing argument (Ssec:sewing).
Deletion ratio and the first-hit comparison.:
-Let $D$ be either a finite cylinder between flat caps, or a finite
+Let D be either a finite cylinder between flat caps, or a finite
convex tiled polygon in the plane. Write
Q_D=
@@ -2175,31 +2164,31 @@ Z_H(sqrt2),&D cylinder,
1,&D plane,
-the positive essential-polygon partition. For a set $K$ of occupied
+the positive essential-polygon partition. For a set K of occupied
centres:
Θ_D(K)=Q_{D∖ K}/Q_D.
[Deletion monotonicity]
-$Θ_D(K)$ is increasing in both the set of available polygons and
-the deleted set $K$, for $0le tlesqrt2$.
+Θ_D(K) is increasing in both the set of available polygons and
+the deleted set K, for 0le tlesqrt2.
Proof.
-For a single deleted polygon $v$:
+For a single deleted polygon v:
Q_P/Q_{P∖ v}=1-tkappa^{|v|}cdot Q_{P∖ N[v]}/Q_{P∖ v},
-where $N[v]$ is the closed conflict neighbourhood. By positivity
-(eq:Q-def) the ratio is $ge1$. Induction on $|P|$ (deleting
+where N[v] is the closed conflict neighbourhood. By positivity
+(eq:Q-def) the ratio is ge1. Induction on |P| (deleting
neighbours successively in the smaller set) gives the multi-deletion
inequality; extra neighbours only increase the ratio.
-For a simple polygon $Lsubset D$ and a boundary source $a$, let
-$P(a,L)$ be the real part of the normalized signed mass of stems that
-first hit $L$. Contour identity:
+For a simple polygon Lsubset D and a boundary source a, let
+P(a,L) be the real part of the normalized signed mass of stems that
+first hit L. Contour identity:
1ge P(a,L)ge
sum_{substack{gamma:ato∂ D gammacap Lne∅}}
@@ -2209,9 +2198,9 @@ kappa^{|gamma|}cos(beta W(gamma))H_D(gamma)ge0.
Boundary-flux primitive.:
[Primitive of boundary flux]
-Choose a dual-edge chain $Gamma$ from a boundary lattice vertex $y_0$
-to an interior vertex $y$. Define $cr_Gamma(gamma)$ to
-be the signed number of $Gamma$-crossings of a completed path and put
+Choose a dual-edge chain Gamma from a boundary lattice vertex y_0
+to an interior vertex y. Define cr_Gamma(gamma) to
+be the signed number of Gamma-crossings of a completed path and put
I=Imsum_{ain∂ D}sum_{gamma:ato∂ D}
cr_Gamma(gamma),kappa^{|gamma|}e^{ibeta W(gamma)}H_D(gamma).
@@ -2225,20 +2214,20 @@ The essential sum is absent in the plane.
Proof. [Sketch]
-Evaluate the same prefix current $J$ two ways.
-(i) Telescoping along $Gamma$ via the cut-link identity of
-Prop.~prop:occupancy: $J=-(mathcal Y(y)/Q_D-1)/calY(y)/Q_D-1)/sqrt2$.
+Evaluate the same prefix current J two ways.
+(i) Telescoping along Gamma via the cut-link identity of
+Prop.~prop:occupancy: J=-(mathcal Y(y)/Q_D-1)/calY(y)/Q_D-1)/sqrt2.
(ii) Continuing each prefix to its first exit gives
-$J=I+2A+E$, where $A$ is the contractible enclosing sum and $|E|le
-sqrt2sum_{essential}kappa^{|L|}H_D(L)sum_aP(a,L)$ (turn
-increments $pm4pi/3$ versus $pm2pi/3$). Comparing gives the
+J=I+2A+E, where A is the contractible enclosing sum and |E|le
+sqrt2sum_{essential}kappa^{|L|}H_D(L)sum_aP(a,L) (turn
+increments pm4pi/3 versus pm2pi/3). Comparing gives the
inequality.
The annular bound.:
[Planar annular bound]
-For every $sge1$ and every interior lattice point $y$:
+For every sge1 and every interior lattice point y:
sum_{substack{L contractible enclosing y
diam(L)in[s,2s]}}kappa^{|L|}le C.
@@ -2246,58 +2235,58 @@ diam(L)in[s,2s]}}kappa^{|L|}le C.
Proof. [Sketch]
-Take a regular lattice hexagon of inradius $asymp s$ around $y$. Only
-chords that separate $y$ from $y_0$ contribute to $I$, with bounded net
+Take a regular lattice hexagon of inradius asymp s around y. Only
+chords that separate y from y_0 contribute to I, with bounded net
crossing; summing the boundary diameter bound over the sources gives
-$|I|le Cs^{3/4}$.
+|I|le Cs^{3/4}.
-A polygon enclosing $y$ with diameter in $[s,2s]$ fits inside a
-doubled hexagon and has transverse span $ge cs$ in some pure direction.
-For order $s$ sources on the corresponding cap, transverse tubes of
-width $ll s$ force intersection of every confined path with $L$, so
+A polygon enclosing y with diameter in [s,2s] fits inside a
+doubled hexagon and has transverse span ge cs in some pure direction.
+For order s sources on the corresponding cap, transverse tubes of
+width ll s force intersection of every confined path with L, so
(eq:channel-bridge-mass) and (eq:first-hit-comparison) give
-$sum_aP(a,L)ge cs^{3/4}$. The essential sum vanishes in the plane; the
+sum_aP(a,L)ge cs^{3/4}. The essential sum vanishes in the plane; the
auxlemma gives the bound.
Bounded cost of large specified lifts.:
[Bounded cost of large lifts]
-Fix $c_2>0$. On a cylinder of period $N$, the total individual critical
-weight of contractible polygons enclosing a fixed mark $y$ whose
-specified lifts have diameter greater than $c_2N$ is at most $C_{c_2}$,
-independently of $N$.
+Fix c_2>0. On a cylinder of period N, the total individual critical
+weight of contractible polygons enclosing a fixed mark y whose
+specified lifts have diameter greater than c_2N is at most C_{c_2},
+independently of N.
Proof. [Sketch]
-Good heights. Put $F_H(t)=-log Z_H(t)$. Superadditivity of
-$F_H$ in $H$ plus the free-energy bound (eq:cylinder-free-energy)
+Good heights. Put F_H(t)=-log Z_H(t). Superadditivity of
+F_H in H plus the free-energy bound (eq:cylinder-free-energy)
make
-a_j=frac{NF_{H_j}(sqrt2)}{H_j},qquad H_j=d_0N2^j,
+a_j=frac{NF_{H_j}(sqrt2)}{H_j}, H_j=d_0N2^j,
-a nondecreasing sequence in $[0,C]$. For any $u>0$, at most $4rC/u$
-indices (for $k=2^r$ a fixed dyadic integer depending on $u$) violate
-$F_{H_j}(sqrt2)-kF_{H_j/k}(sqrt2)le uH_j/(4N)$, and then
+a nondecreasing sequence in [0,C]. For any u>0, at most 4rC/u
+indices (for k=2^r a fixed dyadic integer depending on u) violate
+F_{H_j}(sqrt2)-kF_{H_j/k}(sqrt2)le uH_j/(4N), and then
sum_{L essential in H_j-slab}kappa^{|L|}H_{slab}(L)
le e^{uH_j/N}.
-Good-height contractible sums. For a good height $H$,
+Good-height contractible sums. For a good height H,
sum_{substack{L contractible enclosing y
span_{vert}in[H/4,H/2]}}kappa^{|L|}le Ce^{-cH/N},
-by binning their minimum levels into windows of width $delta N$ (bounded
+by binning their minimum levels into windows of width delta N (bounded
number of bins by good-height hypothesis), applying
(eq:flux-primitive-bound) in an encompassing slab, and using
(eq:cap-first-hit) and the slab-bound auxref{prop:cylinder-barriers}
to control both sides.
Tilted cylinder projection. Dyadic bands with vertical span
-$ge cN$ project injectively onto a cylinder tilted by $60^circ$, of
-period $asymp$ span. The tilted cylinder has its own dyadic-height
+ge cN project injectively onto a cylinder tilted by 60^circ, of
+period asymp span. The tilted cylinder has its own dyadic-height
decomposition; by the above, all but boundedly many heights are good, so
sum_{substack{L contractible enclosing y
@@ -2311,10 +2300,10 @@ diameter intervals.
Classwise convergence of the normalization.:
[Classwise convergence]
-For $rin{0,ipi/2}$ put $sigma(r)=3/8+2r^2/(3pi^2)$ and
-$mathcal A_m(r)=e^calA_m(r)=e^{-Im^2}m^{-sigma(r)}P_m(e^r)$. There exist a fixed
-integer $Qge1$, a $delta>0$, and positive constants
-$mathfrak a_b(r)$,fraka_b(r)$, $0le b<Q$, such that
+For rin{0,ipi/2} put sigma(r)=3/8+2r^2/(3pi^2) and
+mathcal A_m(r)=e^calA_m(r)=e^{-Im^2}m^{-sigma(r)}P_m(e^r). There exist a fixed
+integer Qge1, a delta>0, and positive constants
+mathfrak a_b(r),fraka_b(r), 0le b<Q, such that
mathcal A_m(r)=calA_m(r)=mathfrak a_fraka_{mbmod Q}(r)+O(m^{-delta}).
@@ -2326,33 +2315,33 @@ mathcal Z_calZ_{m,m}(1)=P_m(1)/P_m(i)asymp m^{1/6}.
Proof. [Sketch]
Work with the periodic grand sum at period
-$T=2a_0^{-1}log m+g+itheta$, $|theta|lepi/2$, for a suitably large
-fixed $g$. The stability estimate (eq:Z-coercive) and
+T=2a_0^{-1}log m+g+itheta, |theta|lepi/2, for a suitably large
+fixed g. The stability estimate (eq:Z-coercive) and
localization (eq:gas-localization) continue to hold with the
-coercivity error $O(1/g)mathcal Q$calQ$ absorbed by $g$ large; the period
-graph is fixed-mod-$g$ independent of $m$ on half intervals.
+coercivity error O(1/g)mathcal QcalQ absorbed by g large; the period
+graph is fixed-mod-g independent of m on half intervals.
Multiplying the lifted polynomial by
-$e^{-2Am^2e^{-2T}}e^{-Im^2}m^{-sigma(r)}$ produces an
-$F_m(theta,r)$ whose circle mean is exactly $mathcal A_m(r)$.calA_m(r)$. We show
-$F_m$ is Cauchy at exponential speed along each class modulo $Q$.
+e^{-2Am^2e^{-2T}}e^{-Im^2}m^{-sigma(r)} produces an
+F_m(theta,r) whose circle mean is exactly mathcal A_m(r).calA_m(r). We show
+F_m is Cauchy at exponential speed along each class modulo Q.
Scalar factor. The quadratic bulk contribution gives
-$m^{3/8}$; the logarithmic determinant gives
-$pi T/(16L)=(5)/(24)log m$ plus constant; combining with the
-edge integrals gives $m^{sigma(r)}$ exactly.
+m^{3/8}; the logarithmic determinant gives
+pi T/(16L)=(5)/(24)log m plus constant; combining with the
+edge integrals gives m^{sigma(r)} exactly.
-Edge integrals. Retain only particles within $ε T'$ of an
-edge, at most $T'$ particles, charge $j$ matching the right edge. The
-omitted mass is $e^{-cT'}$, and standard small-slope deformation gives
-agreement of the two retained integrands up to $e^{-cT'}$.
+Edge integrals. Retain only particles within ε T' of an
+edge, at most T' particles, charge j matching the right edge. The
+omitted mass is e^{-cT'}, and standard small-slope deformation gives
+agreement of the two retained integrands up to e^{-cT'}.
-Discrete phases. The $m$-dependent phase factors are roots of
+Discrete phases. The m-dependent phase factors are roots of
unity from fixed logarithmic branch multiplies; the entire phase
-depends only on $mbmod Q$ for a suitable finite $Q$ chosen as a common
+depends only on mbmod Q for a suitable finite Q chosen as a common
multiple of the (finitely many) root-of-unity denominators. The
-retained count is at most $T'$, so $|F_{m'}-F_m|$ is
-$lesssim e^{-cT'}$ for $m'=m+Q$. This proves
+retained count is at most T', so |F_{m'}-F_m| is
+lesssim e^{-cT'} for m'=m+Q. This proves
(eq:classwise-convergence).
@@ -2362,7 +2351,7 @@ Uniform correlations of two prescribed edges.:
We prove the third main theorem: the two-edge marked-polygon bound
(eq:marked-uniform), uniform both in the staircase direction and
-in the ratio $N/d$. The argument has three parts: (a) identify the
+in the ratio N/d. The argument has three parts: (a) identify the
two-edge observable as a contraction of polynomial columns, (b) in the
balanced (antipodal) case reduce it to a two-line gas and estimate,
(c) for unequal intervals control the two long columns outside a small
@@ -2370,28 +2359,28 @@ core.
Setup.:
-Keep the triangular-lattice directions $e_1,e_2$ and identical
-translated marks $E,E+P$ with $P=P_1e_1+P_2e_2$, $P_1,P_2>0$. Thus
-$d=P_1+P_2$ is even, the cylinder period is $kP$ with $kge2$, and
-$N=kd$. The polygon weight is $kappa^{#triangle centres visited}$.
+Keep the triangular-lattice directions e_1,e_2 and identical
+translated marks E,E+P with P=P_1e_1+P_2e_2, P_1,P_2>0. Thus
+d=P_1+P_2 is even, the cylinder period is kP with kge2, and
+N=kd. The polygon weight is kappa^{#triangle centres visited}.
The two restrictions (edge plus displacement) place both marks on a
-monotone staircase of period $P$. The parity of $d$ makes both
+monotone staircase of period P. The parity of d makes both
intervals between the marks even in staircase steps.
Finite marked contraction.:
-Fix integers $kge1$, $0le l,hle k$, and $m,nge0$. Take two lists
-of common slots $A$ (size $m$) and $B$ (size $n$), and four extra lists
-$E_b,H_b$ (sizes $l,h$) and $E_t,H_t$ (sizes $k-l,k-h$). Columns below
-have total length $m+n+l+h$, above $m+n+2k-l-h$.
+Fix integers kge1, 0le l,hle k, and m,nge0. Take two lists
+of common slots A (size m) and B (size n), and four extra lists
+E_b,H_b (sizes l,h) and E_t,H_t (sizes k-l,k-h). Columns below
+have total length m+n+l+h, above m+n+2k-l-h.
[Finite marked contraction]
-For $deltane0$, the contraction
+For deltane0, the contraction
Y=big<Psi_bbig|,delta^{sum_A p_j}big|Psi_tbig>,
of the columns (below with physical rapidities, above with negated
-rapidities, spin-fixed extra lists $-,+,+,-$) satisfies
+rapidities, spin-fixed extra lists -,+,+,-) satisfies
Y=Q_{lh},mathcal J_calJ_{rm ext}!!
sum_{substack{a_jin{-1,0,1}
@@ -2402,37 +2391,36 @@ where the right-hand data are specified below.
Here:
-
-item $Q_{lh}=tfrac12C^{k/2}s^{-k}(-s^{-1})^ls^{-h}$.
-item $J(u)=cos(u+L)cos(u+2L)$, and $mathcal J_calJ_{rm ext}$ is a
-product of $J$-factors between extra slots in each list.
-item $mathcal W(a)=calW(a)=prod_jc_{a_j}prod_{j<j'}P_{a_ja_{j'}}(v_{j'}-v_j)
-prod_{j<j',,same list}D(v_{j'}-v_j)$, with $P$, $D$ from
+ Q_{lh}=tfrac12C^{k/2}s^{-k}(-s^{-1})^ls^{-h}.
+ J(u)=cos(u+L)cos(u+2L), and mathcal J_calJ_{rm ext} is a
+product of J-factors between extra slots in each list.
+ mathcal W(a)=calW(a)=prod_jc_{a_j}prod_{j<j'}P_{a_ja_{j'}}(v_{j'}-v_j)
+prod_{j<j',,same list}D(v_{j'}-v_j), with P, D from
Section~c6:periodic-polynomial.
-item $M^1_+(u)=(1,cos(u+L),J(u))$, $M^1_0=1$ extracted by
-convention, and $M^2_a(u)=M^1_{-a}(-u)$.
-item $mathcal M(a)=calM(a)=prod_{jin Acup B}prod_{einextras}
-M^{d(j,e)}_{a_j}(x_e-v_j)$ with $d(j,e)in{1,2}$ determined by list
+ M^1_+(u)=(1,cos(u+L),J(u)), M^1_0=1 extracted by
+convention, and M^2_a(u)=M^1_{-a}(-u).
+ mathcal M(a)=calM(a)=prod_{jin Acup B}prod_{einextras}
+M^{d(j,e)}_{a_j}(x_e-v_j) with d(j,e)in{1,2} determined by list
and spin type.
Proof. [Sketch]
-Divide both sides by the within-extra-list $J$-products; the resulting
+Divide both sides by the within-extra-list J-products; the resulting
polynomial is separately symmetric in each list. Compare cap
specializations of adjacency: at each extra-common pair, one
-specialization at difference $b$ either eliminates both slots or
+specialization at difference b either eliminates both slots or
converts a common slot on the opposite side into an extra of opposite
-spin, preserving $k$. The scalar fusion identities
-$J(u-b)=J(-u)$ and $D(u)=J(u)J(-u)$ close the recursion. Degrees
+spin, preserving k. The scalar fusion identities
+J(u-b)=J(-u) and D(u)=J(u)J(-u) close the recursion. Degrees
after division match on both sides with definite parities, leaving
-only a constant factor, fixed by the base case $m+n=0$ (equivalently
+only a constant factor, fixed by the base case m+n=0 (equivalently
by saturation of the two fixed-spin columns).
Physical interpretation.:
[Physical two-edge mass]
-For $k=2$, $l=h=1$, $delta=i$, and physical rapidities zero:
+For k=2, l=h=1, delta=i, and physical rapidities zero:
frac{Y}{f_{m+n+2}(0)^2}
@@ -2441,8 +2429,8 @@ prescribed row edges on the cylinder, counted once per polygon.
Proof. [Sketch]
-At $k=2$, $l=h=1$ and $delta=i$, the extra fixed spins at $E$, $H$
-force two oriented arcs from $E$ to $H$, one departing upward and one
+At k=2, l=h=1 and delta=i, the extra fixed spins at E, H
+force two oriented arcs from E to H, one departing upward and one
departing downward, uniquely determined by the polygon's local side
at the marks. Two arcs form the polygon in one orientation only.
The relative signs (from turns, seams, and the phase insertion) yield
@@ -2451,11 +2439,11 @@ a positive weight; the identity extends by height exhaustion.
The antipodal case.:
-When the two marks are antipodal (i.e. $P_1=P_2=m+1$ and $k=2$), the
+When the two marks are antipodal (i.e. P_1=P_2=m+1 and k=2), the
two-line gas of Section~c6:gas directly estimates the marked mass.
[Antipodal bound]
-On the homogeneous cylinder of even period $2m+2$ with antipodal row
+On the homogeneous cylinder of even period 2m+2 with antipodal row
sites, the two-edge marked mass satisfies
K_2le(m+1)^{-4/3+o(1)}.
@@ -2463,25 +2451,25 @@ K_2le(m+1)^{-4/3+o(1)}.
Proof. [Sketch]
-The two extra slots $(E,H)$ supply $J(v)^2$ at an unlowered common
-site plus additional factors $cos^2z/cos^2(z+b)$ per lowering, which
+The two extra slots (E,H) supply J(v)^2 at an unlowered common
+site plus additional factors cos^2z/cos^2(z+b) per lowering, which
amount to a shift of the magnetic potential by a bounded factor. Normalize
-by $Q_{11}C^mD(0)^{m(m-1)}J(0)^{4m}$; this gives the constant
-$P'_m/P_M(i)$ with $M=m+1$, exactly the ratio used in Section~c6:gas.
+by Q_{11}C^mD(0)^{m(m-1)}J(0)^{4m}; this gives the constant
+P'_m/P_M(i) with M=m+1, exactly the ratio used in Section~c6:gas.
The gas estimate at this shifted potential gives the claimed bound.
The magnetic test potential.
-In the residue lift, using the periodized variable $w=ilog$, the
-elementary potential at $z=iw$ is $e^{MV(w)}[-sin(z+L)/sin(z-L)]$.
+In the residue lift, using the periodized variable w=ilog, the
+elementary potential at z=iw is e^{MV(w)}[-sin(z+L)/sin(z-L)].
The bracket is replaced by
U_T(w)=-(S_T(w-iL))/(S_T(w+iL))
(e^{-3iL}(S_T(w-T/2+iL))/(S_T(w-T/2+4iL)))^2,
-which is $T$-periodic, tends to the original bracket, is analytic in
-$e^{-T}$ near zero, and has a zero at $w=iL$ restoring pole
-multiplicity $m$. The first ratio's continuous log increases by
-$12iL$ over a period, compensated by the edge ratio's $-12iL$.
+which is T-periodic, tends to the original bracket, is analytic in
+e^{-T} near zero, and has a zero at w=iL restoring pole
+multiplicity m. The first ratio's continuous log increases by
+12iL over a period, compensated by the edge ratio's -12iL.
@@ -2492,72 +2480,72 @@ Staircases and the two background lists.:
We now prove the uniform two-edge bound (eq:marked-uniform).
The insertion identity (eq:finite-marked-contraction) is unchanged,
but its two lists now have different lengths. The argument keeps two
-scales apart: a logarithmic core of size $asymplog d$ and a full
-period of size $asymplog N$. All constants below are independent of
-$P_1,P_2,k$.
+scales apart: a logarithmic core of size asymplog d and a full
+period of size asymplog N. All constants below are independent of
+P_1,P_2,k.
The staircase row cut.: Use a periodic monotone staircase
-in $e_1,e_2$ (period $P$ through each mark), and translate it in the
+in e_1,e_2 (period P through each mark), and translate it in the
third lattice-edge direction to make rows. This tiles the plane by
lozenges; reading each lozenge as a square tile, the honeycomb path
-weights are the local $R$-matrices $R(t+v_i)$ at argument $L$ or $2L$,
+weights are the local R-matrices R(t+v_i) at argument L or 2L,
the two choices corresponding to an acute or an obtuse angle at the
-corner (they exchange $a,h$ and $c,d$ in the tile notation). After a
+corner (they exchange a,h and c,d in the tile notation). After a
lattice reflection if necessary, take the majority of steps at
-argument $2L$.
+argument 2L.
Through-lines disappear.: The projection argument for two
-marks applies verbatim to the column parameter $t$ above (with its
+marks applies verbatim to the column parameter t above (with its
negative below by crossing). On the lifted slab bounded by two
stairs, the contour from a fixed source has a rotated real coefficient
bounded below positively. Indeed, writing the domain in lattice
-coordinates $(X,Y)$ along $e_1,e_2$ with transverse step $e_2-e_1$,
+coordinates (X,Y) along e_1,e_2 with transverse step e_2-e_1,
the counterclockwise tangent directions from a fixed mid-edge source
-stay within a lower block $0,1$ (and possibly $2$) followed by an
-upper block $3,4$ (and possibly $5$), in units of $pi/3$; after
-subtracting the boundary-closing turn $pi$, the cyclic tangent
-increase lies in $[-pi/3,2pi]$ or in $[0,2pi+pi/3]$, and
-$(3/8)(2pi+pi/3)<pi$ gives the positivity. For the through-mass
-at height $r$, take a lattice hexagon of radius $asymp r$ not
-reaching the far stair, average over $r/N$ translations by whole
+stay within a lower block 0,1 (and possibly 2) followed by an
+upper block 3,4 (and possibly 5), in units of pi/3; after
+subtracting the boundary-closing turn pi, the cyclic tangent
+increase lies in [-pi/3,2pi] or in [0,2pi+pi/3], and
+(3/8)(2pi+pi/3)<pi gives the positivity. For the through-mass
+at height r, take a lattice hexagon of radius asymp r not
+reaching the far stair, average over r/N translations by whole
periods to place the starts in the central part of a common smaller
hexagon, and reverse the exits at the pure sides; the mass is
-$O_N(r^{-1/4})$, so through-lines have vanishing weight and empty
+O_N(r^{-1/4}), so through-lines have vanishing weight and empty
projections retain the same limits.
The two-column picture.: Both extra rapidities in each
-column now take a single value $x_e$. The two common lists have
-sizes $m=d-1$ and $n=N-d-1$. Augment each for the plasma calculation
-by one site at $x_e$ to obtain background lists of sizes $M_A=d$ and
-$M_B=N-d$, both even, with identical proportions of the two possible
-values. Put $lambda_b=pi/(2b)$; let $lambda$ be the proportion of
-minority steps, $0<lambdale1/2$. After a rapidity translation, put
-the majority value at $r_0 L$ and the minority at $(r_0-1)L$, with
-$r_0=1/4+lambda/2$.
+column now take a single value x_e. The two common lists have
+sizes m=d-1 and n=N-d-1. Augment each for the plasma calculation
+by one site at x_e to obtain background lists of sizes M_A=d and
+M_B=N-d, both even, with identical proportions of the two possible
+values. Put lambda_b=pi/(2b); let lambda be the proportion of
+minority steps, 0<lambdale1/2. After a rapidity translation, put
+the majority value at r_0 L and the minority at (r_0-1)L, with
+r_0=1/4+lambda/2.
The residue sum.: As in the balanced two-leg case, the
-potentials are one copy of $R_{gh}(i(w+iv_e))$ per augmented-list
-site of color $h$ and value $v_e$, together with the common magnetic
-factor $-sinh(w+ix_e-iL)/sinh(w+ix_e+iL)$; this follows by shifting
-each extra-slot factor of the balanced calculation. Let $P'$ be the
-remaining residue sum. Using $D(y)=J(y)J(-y)$ and $Q_{11}=C/2$, the
-prefactor divided by $f_N^2$ in the observable is
+potentials are one copy of R_{gh}(i(w+iv_e)) per augmented-list
+site of color h and value v_e, together with the common magnetic
+factor -sinh(w+ix_e-iL)/sinh(w+ix_e+iL); this follows by shifting
+each extra-slot factor of the balanced calculation. Let P' be the
+remaining residue sum. Using D(y)=J(y)J(-y) and Q_{11}=C/2, the
+prefactor divided by f_N^2 in the observable is
frac{C^{N/2}prod_{e<f:,g_e=g_f}D(v_e-v_f)}{2f_N^2}
;prod_{e common}(J(v_e-x_e))/(J(x_e-v_e)).
-Lowering counts satisfy $K_g=M_g+u$ with a common integer $u$.
+Lowering counts satisfy K_g=M_g+u with a common integer u.
Periodize with the magnetic factor
[
U_T(w)=-(S_T(w+ix_e-iL))/(S_T(w+ix_e+iL))
(e^{-3iL}(S_T(w-T/2+iL))/(S_T(w-T/2+4iL)))^2,
]
-and put background charges $q_e=-q_{g_e}$ at offsets $a_e=-v_e$ in
-the central theta sum; then the Gaussian center $X$ includes
-$sum_e q_e i a_e$ and periodic differentials are recovered. The
+and put background charges q_e=-q_{g_e} at offsets a_e=-v_e in
+the central theta sum; then the Gaussian center X includes
+sum_e q_e i a_e and periodic differentials are recovered. The
nome limit and rapidity continuation work as in the balanced case,
-with singularities at differences exactly $L$ removed by continuation
+with singularities at differences exactly L removed by continuation
in rapidities.
The ordered magnetic gas.:
@@ -2569,15 +2557,13 @@ winding. Phases are retained in this step: the interval estimate
below uses the resulting transfer operator rather than the modulus
of an unidentified periodic partition.
-The extended mode matrix.: Put $E(p)=sum_e e^{-p a_e}{bf e}_{g_e}$.
-Include extra background slots $E_e$ of count-mode 1 in the mode
-matrix, with pair entries from the potentials (${cal M}_{R,E}n_E=-Kwidehatrho E$)
+The extended mode matrix.: Put E(p)=sum_e e^{-p a_e}{bf e}_{g_e}.
+Include extra background slots E_e of count-mode 1 in the mode
+matrix, with pair entries from the potentials ({cal M}_{R,E}n_E=-Kwidehatrho E)
and inverse couplings initially zero; their mutual block is zero
-except at Laurent zero, where we assign $4q_eq_f/p^2+4q_eq_f(a_e-a_f)/p$
+except at Laurent zero, where we assign 4q_eq_f/p^2+4q_eq_f(a_e-a_f)/p
with zero finite part. Then:
-
-
-item The bulk log term is
+ The bulk log term is
S(T)=frac1{2T}sum_p E(-p)^tK(p)E(p)widehatrho(p)^2
-tfrac12(T/3-2log2)(M_A-M_B)^2.
@@ -2585,86 +2571,85 @@ S(T)=frac1{2T}sum_p E(-p)^tK(p)E(p)widehatrho(p)^2
Determinants and measure conventions of (eq:legacy-P-star) are
unchanged.
-
-item The chemical potentials per final particle (before the magnetic
+ The chemical potentials per final particle (before the magnetic
test) are
2pi i,sigma_i!!!sum_{e:,g_e=g_i}!!!
int_0^{s+i(v_e+delta_i L)}!!!
-sum_{nuinmathbb Z}bbZrho(z+nu T),dz,quad
+sum_{nuinmathbb Z}bbZrho(z+nu T),dz,
sigma=(0,1,1,-1,0).
-Fugacity still brings $Tr_j^2/(4h_*)-r_j(u_*+barbar L)$.L)$.
+Fugacity still brings Tr_j^2/(4h_*)-r_j(u_*+barbar L).L).
Extended zero modes.: At zero, take the raw right count
vector with left and inverse modes as before, put the background
-modes at 1, and prescribe $q^tgamma'_{rm raw}
-=A_L+sum_e q_ea_e+pi l/2$. The extended matrix has the same
-charge/offset/winding poles, and the saddle term $widehatrho E$
+modes at 1, and prescribe q^tgamma'_{rm raw}
+=A_L+sum_e q_ea_e+pi l/2. The extended matrix has the same
+charge/offset/winding poles, and the saddle term widehatrho E
neutralizes the appended charges at 0 and contributes exactly
-$sum_e q_ea_e$ to the derivative condition, while its count value
-$(M_A,M_B)$ leaves the free integer and Gaussian shift unchanged. The
+sum_e q_ea_e to the derivative condition, while its count value
+(M_A,M_B) leaves the free integer and Gaussian shift unchanged. The
assigned background block has zero contracted finite part at zero, and
the linear symbol at a particle against a same-color site is
-$-2pisigma_i e^{(v_e+delta_i L)p}widehatrho(p)/p$; the primitive
+-2pisigma_i e^{(v_e+delta_i L)p}widehatrho(p)/p; the primitive
of (eq:legacy-P-25) follows.
Ordered-position form.: For particles in increasing real
-contour order write $H(s)=j+sum_{Re s_d>Re s}W_d$ ($j$ the Poisson
-integer), with mid-value $H_m$ at jumps; winding neutrality closes the
-integer path. Set $c^t$ by
+contour order write H(s)=j+sum_{Re s_d>Re s}W_d (j the Poisson
+integer), with mid-value H_m at jumps; winding neutrality closes the
+integer path. Set c^t by
[
{bf1}^t(J(p)-K(p)ell)=2pi W^t/p-2h_*c^t+O(p),
]
-so $u_*+barbar L=c^tn-(L=c^tn-(pi i/h_*)sum W_ds_d$. (Here $J$ is the
+so u_*+barbar L=c^tn-(L=c^tn-(pi i/h_*)sum W_ds_d. (Here J is the
coupling block of the transform, not the site function of
(eq:legacy-P-23).) At zero,
-Z=alpha WW^t/p^2+M_*/p+O(1),qquad
+Z=alpha WW^t/p^2+M_*/p+O(1),
M_*=-2pi(cal Wtilde+Wc^t-cW^t),
-where $cal Wtilde$ is an integral skew matrix: change right
-variables by $-ell n$ before Schur, then the non-right block has
-simple coefficient $-2pical Wtilde$ from the shifted winding
+where cal Wtilde is an integral skew matrix: change right
+variables by -ell n before Schur, then the non-right block has
+simple coefficient -2pical Wtilde from the shifted winding
matrix, and the remaining poles come from the common channel.
Constant phases.: Combined with the ordered factor from
-$M_*/p$ and $-2pi ijc^tn$, the explicit terms of
+M_*/p and -2pi ijc^tn, the explicit terms of
(eq:legacy-P-star) and the site/test couplings reduce, up to a
single unit prefactor, to
prod_deta_{i_d}exp[-2pi i c_{i_d}H_m(s_d)]
-for fixed unit constants $eta$ per species, valid for every choice of
-even $M_A,M_B$ (including 0). The argument uses the $-i/2$ pole
+for fixed unit constants eta per species, valid for every choice of
+even M_A,M_B (including 0). The argument uses the -i/2 pole
kernel at ordered (positive) separation, the unordered sign
-$(-1)^{cal Wtilde_{ab}}$ per pair from log constants, and
+(-1)^{cal Wtilde_{ab}} per pair from log constants, and
residue-orientation and Weyl sign contributions whose second
-differences over count space reproduce $cal Wtilde$; these
-cancel the unordered signs. At every branch change of order $nu$
-(across a singularity) $cal Wtilde$ shifts by the same
-amount and the sign $(-1)^nu$ appears, so (eq:legacy-P-27)
+differences over count space reproduce cal Wtilde; these
+cancel the unordered signs. At every branch change of order nu
+(across a singularity) cal Wtilde shifts by the same
+amount and the sign (-1)^nu appears, so (eq:legacy-P-27)
persists under fusion and exterior branch changes.
The short pair kernel.: For ordered positive separations
-use the periodization of $G_{ab}(s)$ with a regular diagonal at
+use the periodization of G_{ab}(s) with a regular diagonal at
contact, counting both periodic continuations once per positively
separated pair (transpose at negative separations). This is the
inverse Fourier kernel after removing the two pole kernels of
(eq:legacy-P-26); thus the off-contact gas exponent is minus one
-copy of $G$ per such pair (including translated partners), and the
-regular contact self term carries half weight. For $Re s>0$,
+copy of G per such pair (including translated partners), and the
+regular contact self term carries half weight. For Re s>0,
G_{ab}(s)=sum_{lge1}sum_{k=0}^{r_*}C_{ab,l,k}s^ke^{-omega_ls},
-with $omega_l>0$ growing at least linearly and $C_{ab,l,k}$ growing
+with omega_l>0 growing at least linearly and C_{ab,l,k} growing
at most polynomially: the final two-line symbols have only unit-circle
-poles in $e^{Lp}$, of bounded order, so shifting the inverse contour
+poles in e^{Lp}, of bounded order, so shifting the inverse contour
down in imaginary periods yields the residues above the separated
zero-pole kernels, and tail rotation adds an exponentially vanishing
remainder.
@@ -2678,165 +2663,165 @@ control products of intervals with different potentials, so we need a
common operator throughout.
Homogeneous reference configuration.: Fix a cell length
-$S_0$ and consider a homogeneous gas with no magnetic functions or
-(eq:legacy-P-25), but with constant $r=-2a_0'$,
-$a_0'=pm6Li$, all five color-$A$ species allowed, and only $S,F$ of
-color $B$. Beyond the pair term $G$ and the phases
+S_0 and consider a homogeneous gas with no magnetic functions or
+(eq:legacy-P-25), but with constant r=-2a_0',
+a_0'=pm6Li, all five color-A species allowed, and only S,F of
+color B. Beyond the pair term G and the phases
(eq:legacy-P-27), the exponent contains
-h_*^{-1}int(pi H+ia_0')^2,ds+2a_0'c^tn.
-Cell states and gauge.: A cell state $x$ is an incoming
+Cell states and gauge.: A cell state x is an incoming
integer height plus indistinguishable points of each species/color in
-an interval of length $S_0$, with factorial Lebesgue measure. Along
-an edge $xto y$ require outgoing height of $x$ to equal incoming of
-$y$. Use the reference energy at zero tilt and no extra particle
-potential: $A(x,y)$ is half of each self energy (including
-$pi^2/h_*int H^2$) plus the real adjacent pair energy from
+an interval of length S_0, with factorial Lebesgue measure. Along
+an edge xto y require outgoing height of x to equal incoming of
+y. Use the reference energy at zero tilt and no extra particle
+potential: A(x,y) is half of each self energy (including
+pi^2/h_*int H^2) plus the real adjacent pair energy from
(eq:legacy-P-28) (cyclically omitting separations of at least
-two cells). By (eq:legacy-P-P3), for large $S_0$,
+two cells). By (eq:legacy-P-P3), for large S_0,
sum A(x_i,x_{i+1})ge4ε_Ssum|x_i|_0^2-C_Sn_{rm cells},
-qquad|x|_0^2=N_x^2+H_{x,rm in}^2,
+|x|_0^2=N_x^2+H_{x,rm in}^2,
-$ε_S>0$ at least inverse-polynomial in $S_0$. There is a Borel
-$phi$ on states away from coincidences with
+ε_S>0 at least inverse-polynomial in S_0. There is a Borel
+phi on states away from coincidences with
[
A(x,y)ge2ε_S(|x|_0^2+|y|_0^2)-C_S+phi(y)-phi(x),
]
obtained as the infimum of corrected path sums from the empty height
zero state (finite since unit jumps of both signs are available).
-Gauging each forward edge by $exp(phi(y)-phi(x))$ cancels on cycles,
+Gauging each forward edge by exp(phi(y)-phi(x)) cancels on cycles,
splits complex short comparisons into local errors
-$O_S(slope)(N_x^2+N_y^2)$, and leaves Gaussian margins in
+O_S(slope)(N_x^2+N_y^2), and leaves Gaussian margins in
both endpoints, including for small-slope complex cell deformations.
Fock encoding of long-range interactions.: For
interactions of range at least two cells add a bosonic symmetric Fock
space over countably many finite propagation blocks. Use semigroups
-$D_d$ generated by $-omega_l+nilpotent$, with creation and
-detection arrays $u_a,v_b$ satisfying $v_bD_du_a=-G_{ba}(d)$ term by
+D_d generated by -omega_l+nilpotent, with creation and
+detection arrays u_a,v_b satisfying v_bD_du_a=-G_{ba}(d) term by
term in (eq:legacy-P-28). The Fock operator in forward order at
-edge $x_ito x_{i+1}$ is
+edge x_ito x_{i+1} is
[
exp a^dagger(U) Gamma(D_{ell'}) exp a(V),
]
-with $V=sum_{bin x_{i+1}}v_bD_{d_b^L}$,
-$U=sum_{ain x_i}D_{ell'+d_a^R}u_a$, symmetric tensor powers, and
+with V=sum_{bin x_{i+1}}v_bD_{d_b^L},
+U=sum_{ain x_i}D_{ell'+d_a^R}u_a, symmetric tensor powers, and
exponential-vector normalization. Each block is rescaled so
-$|U|+|V|le Ce^{-ε S_0}(N_x+N_y)$; splitting
-$Gamma(D_{ell'})=Gamma(tfrac12I)Gamma(4D_{ell'})Gamma(tfrac12I)$
-and using $< a^dagger(U)^k>le|U|^k√((m+k)!/m!)$
-bounds the trace norm by $exp(C|U|^2+C|V|^2)$, within the
-Gaussian margins. The resulting integral operators on $L^2$ with the
+|U|+|V|le Ce^{-ε S_0}(N_x+N_y); splitting
+Gamma(D_{ell'})=Gamma(tfrac12I)Gamma(4D_{ell'})Gamma(tfrac12I)
+and using < a^dagger(U)^k>le|U|^k√((m+k)!/m!)
+bounds the trace norm by exp(C|U|^2+C|V|^2), within the
+Gaussian margins. The resulting integral operators on L^2 with the
classical state are Hilbert--Schmidt, also under bounded point log
moduli per particle and bounded perturbations of the volume or height
energies.
[Homogeneous interval operators]
On the ambient cell space with the Fock fibers above, let
-$mathcal T_calT_pm$ be homogeneous cell operators with all five color-$A$
-species and only $S,F$ of color $B$, other species having zero
+mathcal T_calT_pm be homogeneous cell operators with all five color-A
+species and only S,F of color B, other species having zero
outgoing point weight, no magnetic functions or potentials
-(eq:legacy-P-25), constant tilt $a_0'=pm6Li$, $r=-2a_0'$, and
-weights consisting of the pair term $G$, the phases
+(eq:legacy-P-25), constant tilt a_0'=pm6Li, r=-2a_0', and
+weights consisting of the pair term G, the phases
(eq:legacy-P-27), and the exponent (eq:legacy-P-29).
Then both are Hilbert--Schmidt and
-$spr(mathcal T_calT_pm)<exp(S_0/2)$.
+spr(mathcal T_calT_pm)<exp(S_0/2).
Proof.
The Hilbert--Schmidt assertion follows from the Fock trace-norm bound.
Cyclic traces. Traces of cyclic products (at least two edges)
-give exactly the ordered periodic gas times $det(I-D_T)^{-1}$, the
-last tending to 1 at large period $T=m_0S_0$. Normal-ordering the
-Fock product as $e^{a^dagger(P)}Gamma(D_T)e^{a(B)}$ plus internal
-contractions, the normalized trace is $exp(B(I-D_T)^{-1}P)$, which
+give exactly the ordered periodic gas times det(I-D_T)^{-1}, the
+last tending to 1 at large period T=m_0S_0. Normal-ordering the
+Fock product as e^{a^dagger(P)}Gamma(D_T)e^{a(B)} plus internal
+contractions, the normalized trace is exp(B(I-D_T)^{-1}P), which
recovers precisely (eq:legacy-P-28) at positive separations of at
least two cells and nothing else.
Spectral radius via powers. For a compact Hilbert--Schmidt
-operator, $spr(mathcal T)=limsup_mcalT)=limsup_m|mathcal T^mcalT^m|^{1/m}$;
+operator, spr(mathcal T)=limsup_mcalT)=limsup_m|mathcal T^mcalT^m|^{1/m};
splitting off finite-dimensional outer eigenvalues leaves a strictly
smaller radius on a complementary invariant space whose power traces
are trace-class. It suffices to bound the homogeneous log-trace per
-unit length along periods $m_0S_0$, $m_0to∞$.
+unit length along periods m_0S_0, m_0to∞.
-Two-line transform. Fix real $T$ large and apply the two-line
-residue transform with no magnetic potential, $r=-2a_0'$, and
-backgrounds $M_A=0,M_Bto∞$ even at zero. The original sum has no
-$A$ particles, but opening and transforming both lines is still valid
+Two-line transform. Fix real T large and apply the two-line
+residue transform with no magnetic potential, r=-2a_0', and
+backgrounds M_A=0,M_Bto∞ even at zero. The original sum has no
+A particles, but opening and transforming both lines is still valid
by the coefficientwise proof of (eq:legacy-P-star); after
removing (eq:legacy-P-24) the transformed gas retains exactly
-the permitted species, because nonzero (eq:legacy-P-25) on $B$
-has real part strictly negative proportional to $M_B$ on this fixed
+the permitted species, because nonzero (eq:legacy-P-25) on B
+has real part strictly negative proportional to M_B on this fixed
real period. The remaining exponent is (eq:legacy-P-29) via
-$int(H-j)=sum W_ds_d$, and its determinant scalar per unit length in
-log is $1/2+o(1)$.
+int(H-j)=sum W_ds_d, and its determinant scalar per unit length in
+log is 1/2+o(1).
One-line comparison. The same residue sum also admits a
-one-right-line calculation on $B$ with heat period $2bc_*$, a single
-insertion, Cartan kernel $K_{0,BB}=(2pi/p)coth(bp)$, and dressing
-$1-e$. A single Weyl reflection twists the oscillator trace by the
+one-right-line calculation on B with heat period 2bc_*, a single
+insertion, Cartan kernel K_{0,BB}=(2pi/p)coth(bp), and dressing
+1-e. A single Weyl reflection twists the oscillator trace by the
oscillator sign, contributing
-$prod_{p>0}(1-e^{-2bp})/(1+e^{-2bp})$ and replacing the reciprocal
-Cartan multiplier by $tanh(bp)$; the pair symbols reduce to same-color
-entries (color entry $c=0$). The scalar (eq:legacy-P-24) is
+prod_{p>0}(1-e^{-2bp})/(1+e^{-2bp}) and replacing the reciprocal
+Cartan multiplier by tanh(bp); the pair symbols reduce to same-color
+entries (color entry c=0). The scalar (eq:legacy-P-24) is
unchanged; (eq:legacy-P-25) applies; other zero factors grow only
-polynomially in $T$.
+polynomially in T.
Real same-color kernel. For positivity put
-$D_0(z)=(z^2-z+1)(z^4-z^2+1)$, $t=e^{x/12}$,
-$H_{ij}=t^{-36+12(delta_i-delta_j)}D_0(t^{12})F_{ij}^{(0)}(t^{12})$
-with $x=Lp>0$. After clearing positive scalars one checks that
-$(H+H^t)/(4sinh x)$ is positive definite. Here $(H+H^t)/2$ is
+D_0(z)=(z^2-z+1)(z^4-z^2+1), t=e^{x/12},
+H_{ij}=t^{-36+12(delta_i-delta_j)}D_0(t^{12})F_{ij}^{(0)}(t^{12})
+with x=Lp>0. After clearing positive scalars one checks that
+(H+H^t)/(4sinh x) is positive definite. Here (H+H^t)/2 is
Laurent of degree at most 36, opposite coefficients negatives of one
-another; each coefficient $B_j$ of $t^j$ ($1le jle36$) enters with
-multiplier $sinh(beta x)/sinh x$, $beta=j/12$. With
-$a=2cosh x-2$ and nodes $1,4,9$ the interpolation coefficients are
+another; each coefficient B_j of t^j (1le jle36) enters with
+multiplier sinh(beta x)/sinh x, beta=j/12. With
+a=2cosh x-2 and nodes 1,4,9 the interpolation coefficients are
[
-u_0=beta,quad u_1=beta(beta^2-1)/6,quad
+u_0=beta, u_1=beta(beta^2-1)/6,
u_2=beta(beta^2-1)(beta^2-4)/120,
]
and the lower polynomial has first two coefficients
-$U_k=sum_ju_kB_j$ ($k=0,1$) and last
+U_k=sum_ju_kB_j (k=0,1) and last
[
U_2=sum_ju_2B_j-tfrac12sum_{12nmid j}
(u_2B_j+|u_2|diag(|B_j|_{rm ent}{bf1})).
]
-A finite integer check (1000 times each $U_k$ is integer-valued with
-consecutive leading determinants positive after subtracting $3I$)
-verifies that $U_0,U_1,U_2$ are positive definite; the rounding error
-has $5times5$ operator norm at most $5/2$, so the exact matrices are
+A finite integer check (1000 times each U_k is integer-valued with
+consecutive leading determinants positive after subtracting 3I)
+verifies that U_0,U_1,U_2 are positive definite; the rounding error
+has 5times5 operator norm at most 5/2, so the exact matrices are
positive as well.
-Final numerical bound. Only $S,F$ survive. Their symmetric
-entries, omitting $2pi/p$, on the diagonal are
-$(sinh3x+sinh xmpsinh2x)/cosh3x$ and off-diagonal
-$(sinh(5x/2)+sinh(x/2))cosh(x/12)/cosh3x$. The spatial real
-kernels are nonnegative off contact (each divided by $p$ is a
-nonnegative mixture of $cosh(hp)/cosh(bp)$, $0le h<b$). The regular
-diagonal is at least $2log L$ (each $F^{(0)}$ diagonal is at least
-$1-e^{-x}$ for positive $x$), improving under periodization, so the
-total real pressure from particles is at most $1/(pi L)$. The
-one-line determinant per length contributes $1/4-1/6+o(1)=1/12+o(1)$.
+Final numerical bound. Only S,F survive. Their symmetric
+entries, omitting 2pi/p, on the diagonal are
+(sinh3x+sinh xmpsinh2x)/cosh3x and off-diagonal
+(sinh(5x/2)+sinh(x/2))cosh(x/12)/cosh3x. The spatial real
+kernels are nonnegative off contact (each divided by p is a
+nonnegative mixture of cosh(hp)/cosh(bp), 0le h<b). The regular
+diagonal is at least 2log L (each F^{(0)} diagonal is at least
+1-e^{-x} for positive x), improving under periodization, so the
+total real pressure from particles is at most 1/(pi L). The
+one-line determinant per length contributes 1/4-1/6+o(1)=1/12+o(1).
The homogeneous log trace per unit length is therefore at most
[
frac1{pi L}+frac1{12}-frac12+o(1)
=frac8{pi^2}-frac5{12}+o(1)<(17)/(36)<frac12
]
-using $L=pi/8$ and $pi>3$. This proves the strict spectral bound for
+using L=pi/8 and pi>3. This proves the strict spectral bound for
either sign of the tilt.
For later use, the spectral-radius formula gives a fixed block length
-$q$ and $ε>0$ with
-$|mathcal T_calT_pm^q|leexp((1/2-ε)qS_0)$ for both tilts.
+q and ε>0 with
+|mathcal T_calT_pm^q|leexp((1/2-ε)qS_0) for both tilts.
Continuity of a finite ordered product in operator norm then preserves
-$exp(qS_0/2)$ for sufficiently close $q$-blocks in the same state
+exp(qS_0/2) for sufficiently close q-blocks in the same state
space and gauge. Section~u:core-product proves this closeness
for the actual tail kernels; the core requires its own estimate.
@@ -2848,13 +2833,13 @@ magnetic source itself, prove the uniform one-particle estimates
needed to compare its tails with that operator, and retain the
suppression inside the shorter core.
-The magnetic exponent.: Write $a(s)=log U_T(s)$ on the
+The magnetic exponent.: Write a(s)=log U_T(s) on the
right contours, in the branch whose bulk log grows from the odd
-primitive in $s+ix_e$; its total winding around the period is zero,
+primitive in s+ix_e; its total winding around the period is zero,
as in the balanced case (Sc6:subheading-16). On the right,
-the mean $barbar a$a$ gives $r=-2barbar a$a$ and scalar $(M_A+M_B)barbar a$;a$; on
-the left retain direct $U_T$ factors with the mean subtracted; on the
-right insert $a-barbar a$.a$. Completing Cartan nonzero modes as in the
+the mean barbar aa gives r=-2barbar aa and scalar (M_A+M_B)barbar a;a; on
+the left retain direct U_T factors with the mean subtracted; on the
+right insert a-barbar a.a. Completing Cartan nonzero modes as in the
balanced case yields an extra external scalar
S_a(T)=frac1Tsum_pwidehatrho(p){bf1}^tE(-p)widehatwidehat a(p)a(p)
@@ -2867,9 +2852,9 @@ and the local height-square becomes
The added particle-source nonzero-mode multiplier is
-$-J(-p)^t{bf1}Q_+(p)$; its simple pole $+2pi W/(ph_*)$ acting on
-$a-barbar a$a$ gives the mean-zero primitive of $-i(a-barbar a)$,a)$, and
-summing the jumps by parts against $H$ recovers the quadratic in
+-J(-p)^t{bf1}Q_+(p); its simple pole +2pi W/(ph_*) acting on
+a-barbar aa gives the mean-zero primitive of -i(a-barbar a),a), and
+summing the jumps by parts against H recovers the quadratic in
(eq:legacy-P-26) together with the mean and test-variance
pieces kept in (eq:legacy-P-31). The remaining per-particle
exponent, beyond (eq:legacy-P-27), is
@@ -2879,41 +2864,40 @@ exponent, beyond (eq:legacy-P-27), is
+b_i'(s),
-where the $b_{i'}$ are the physical integer offsets and $b_i'$ is the
-Fourier action on $a'$ of the regular symbol
-$[-J(-p)^t{bf1}Q_+(p)-2pi W/(ph_*)-(2c-ell_Σ)]_i/(-ip)$.
-Here $Q_+=1/h_*+O(p^2)$ fixes the constant $2c-ell_Σ$, whose
+where the b_{i'} are the physical integer offsets and b_i' is the
+Fourier action on a' of the regular symbol
+[-J(-p)^t{bf1}Q_+(p)-2pi W/(ph_*)-(2c-ell_Σ)]_i/(-ip).
+Here Q_+=1/h_*+O(p^2) fixes the constant 2c-ell_Σ, whose
mean times the counts matches the terms in (eq:legacy-P-31) and
-(eq:legacy-P-32) via $int(H-j)=sum W_ds_d$.
+(eq:legacy-P-32) via int(H-j)=sum W_ds_d.
-Period and bend.: Keep the cell length $S_0$, block length
-$q$, and spectral margin fixed. Choose the bend size $theta_0$ small,
-then transition length $A$ and buffer $ggg S_0$ large enough to meet
+Period and bend.: Keep the cell length S_0, block length
+q, and spectral margin fixed. Choose the bend size theta_0 small,
+then transition length A and buffer ggg S_0 large enough to meet
the slope bounds below; the fixed boundary layers of the product
estimate may depend on those choices, but no constant depends on
-$d,N,lambda$.
-
-Take nominal period $R=2a_0^{-1}log N+g+O(S_0)$, a multiple of
-$2S_0$, and admit $|T-R|<pi+1$. Parameterize positions by
-$s=t+iy(t)+(T-R)chi(t)$, $-R/2le tle R/2$, with $y$ periodic,
-real, Lipschitz small, supported outside $|t|>A$, decreasing to
-$-theta_0L$ by $|t|=2A$, and flat after; here $chi$ is the analytic
-period adjustment of (eq:legacy-P-P3), derivative $O(1/g)$,
-$chi(pm R/2)=pm1/2$, supported within $g/3$ of the ends. Slopes
+d,N,lambda.
+
+Take nominal period R=2a_0^{-1}log N+g+O(S_0), a multiple of
+2S_0, and admit |T-R|<pi+1. Parameterize positions by
+s=t+iy(t)+(T-R)chi(t), -R/2le tle R/2, with y periodic,
+real, Lipschitz small, supported outside |t|>A, decreasing to
+-theta_0L by |t|=2A, and flat after; here chi is the analytic
+period adjustment of (eq:legacy-P-P3), derivative O(1/g),
+chi(pm R/2)=pm1/2, supported within g/3 of the ends. Slopes
satisfy all smallness requirements.
Uniform one-particle estimates.:
-
-item $a(s)$ tends exponentially to $a_pm=pm6Li$ on $pm t>0$
-away from $0$ and the period edges, and is bounded; the logs of
+ a(s) tends exponentially to a_pm=pm6Li on pm t>0
+away from 0 and the period edges, and is bounded; the logs of
direct multipliers in (eq:legacy-P-33) tend to these phases
-modulo $2pi i$ with bounded modulus; and $b_i'$ tends to $0$
-exponentially with $exp(Re b_i')$ bounded throughout.
-item For $D$ and $I_+'$ the real exponent in (eq:legacy-P-25)
-is at most $-c_0M_ge^{-a_0|t|}+O(1)$; the same bound holds for $I_-'$
-when $|t|ge 2A$, and everywhere the exponent is $O(1)$. For color
-$A$, the (eq:legacy-P-25) multipliers tend uniformly to $1$
-outside $|t|le a_0^{-1}log d$ and away from the period edges.
+modulo 2pi i with bounded modulus; and b_i' tends to 0
+exponentially with exp(Re b_i') bounded throughout.
+ For D and I_+' the real exponent in (eq:legacy-P-25)
+is at most -c_0M_ge^{-a_0|t|}+O(1); the same bound holds for I_-'
+when |t|ge 2A, and everywhere the exponent is O(1). For color
+A, the (eq:legacy-P-25) multipliers tend uniformly to 1
+outside |t|le a_0^{-1}log d and away from the period edges.
Proof of 1. The product formula and derivative periodizations
@@ -2922,52 +2906,52 @@ have no poles along the graph (edge compensator unshifted in
rapidity). The short term uses the regular multiplier on the two
whole-line coth difference across one period before periodization; its
transform off zero decays at the nearest shifted singularity. For
-$I_-'$, the isolated core log difference under the shift $idelta_iL$
+I_-', the isolated core log difference under the shift idelta_iL
has bounded modulus above and is exponentially localized; after this
subtraction the extra exponent has uniform frequency decay because
-$A_3=-1$. Contour rotation away from the centers, as in
+A_3=-1. Contour rotation away from the centers, as in
(eq:legacy-P-P3), gives all asserted bounds. Widths on the
unshifted ratios are bounded away from singularities since
-$-3/4le x_e/Lle1/2$.
+-3/4le x_e/Lle1/2.
Proof of 2. On the basic line the imaginary part of a single
-primitive at argument $t+iLY$ is
+primitive at argument t+iLY is
[
B(u,Y)=frac1{2pi}atanh
(4sqrt3cosh usin(pi Y/3))/(3+4sinh^2u+4sin^2(pi Y/3)),
-quad u=a_0t,
+ u=a_0t,
]
by integrating the explicit density. Before bending, mix
-$Y=r_0+delta_i$ and $Y-1$ with proportions $1-lambda$ and $lambda$.
-For $I_+'$ the sign is uniformly positive with size $ge ce^{-|u|}$
-($Yge 2/3$ and the positive function is increasing on $0<Y<1$); for
-$D$ the same holds with $Y=1/3+lambda/2$, by monotonicity and
+Y=r_0+delta_i and Y-1 with proportions 1-lambda and lambda.
+For I_+' the sign is uniformly positive with size ge ce^{-|u|}
+(Yge 2/3 and the positive function is increasing on 0<Y<1); for
+D the same holds with Y=1/3+lambda/2, by monotonicity and
continuity. The remaining cases reduce to bounding
[
(2pi B)/(4sqrt3cosh usin(pi Y/3)/(3+4sinh^2u))
]
-between $1$ and $atanh(sqrt z)/sqrt z$, with
-$z=4sin^2(pi Y/3)/(3+4sinh^2u)$. For $D$ with $lambdale1/3$,
+between 1 and atanh(sqrt z)/sqrt z, with
+z=4sin^2(pi Y/3)/(3+4sinh^2u). For D with lambdale1/3,
it suffices that
[
(1-lambda)sin(pi Y/3)-1.33lambdasin(pi(1-Y)/3)>0;
]
-splitting at $lambda=1/4$ gives $ge17/600$ for
-$0lelambdale1/4$ (via $sin(pi/9)>1/3,sin(2pi/9)<2/3$) and
-$ge337/6000$ for $1/4lelambdale1/3$ (via $sin(11pi/72)>9/20$
-and $sin(13pi/72)<11/20$), using
-$atanh(sqrt z)/sqrt z<133/100$ for $zle5/9$ (from
-the series $1+z/3+z^2/5+z^3/(7(1-z))$). For $I_-'$,
-$Y=lambda/2le1/4$, so $z<1/10$ and
-$atanh(sqrt z)/sqrt z<11/10$; then
-$(11/10)(1-lambda)sin(pilambda/6)le(11pi/60)lambda
-<lambdasin(pi/4)lelambdasin(pi(1-Y)/3)$, giving the negative
-sign with margin $lambda e^{-|u|}$. Bending improves these
-comparisons for small $theta_0$; the primitive limits to $pm1/2$
+splitting at lambda=1/4 gives ge17/600 for
+0lelambdale1/4 (via sin(pi/9)>1/3,sin(2pi/9)<2/3) and
+ge337/6000 for 1/4lelambdale1/3 (via sin(11pi/72)>9/20
+and sin(13pi/72)<11/20), using
+atanh(sqrt z)/sqrt z<133/100 for zle5/9 (from
+the series 1+z/3+z^2/5+z^3/(7(1-z))). For I_-',
+Y=lambda/2le1/4, so z<1/10 and
+atanh(sqrt z)/sqrt z<11/10; then
+(11/10)(1-lambda)sin(pilambda/6)le(11pi/60)lambda
+<lambdasin(pi/4)lelambdasin(pi(1-Y)/3), giving the negative
+sign with margin lambda e^{-|u|}. Bending improves these
+comparisons for small theta_0; the primitive limits to pm1/2
along the real axis and bends add exponentially small errors.
Continuation to the magnetic graph.: The mode transform is
-justified by analytic continuation. Start at real large $T$ near
+justified by analytic continuation. Start at real large T near
zero rapidities, where the trace and final fusions are pole-free (as
in the balanced case). Continue rapidities linearly to the targets
on the final contours with horizontal projections; absolute
@@ -2978,7 +2962,7 @@ fictitious background only for actual remaining residue sites, giving
meromorphic continuation with a finite product-sum representation;
the straight continuation removes collision-divisor singularities on
the used rapidity family. Deform the position graphs to the small
-$y$ on real periods by simultaneous transport. Coincident loci are
+y on real periods by simultaneous transport. Coincident loci are
zeros of nonnegative integer order (analytic transport). This yields
holomorphic continuation in the period disk on the chosen
parametrization, with remaining split-residue singularities on fixed
@@ -2989,24 +2973,24 @@ equal to the original trigonometric sum at nome zero.
A bounded product through the core and tails.:
-Choose $u_0ge0$ a cell-boundary coordinate within bounded distance of
-$a_0^{-1}log d$. On the core $[-u_0,u_0]$, where $chi=0$, remove
-the scalar $h_*^{-1}int a^2$ from (eq:legacy-P-32), of real part
-$-2(2u_0)+O(1)$ since $(6L)^2/h_*=2$. Redistribute the principal real
+Choose u_0ge0 a cell-boundary coordinate within bounded distance of
+a_0^{-1}log d. On the core [-u_0,u_0], where chi=0, remove
+the scalar h_*^{-1}int a^2 from (eq:legacy-P-32), of real part
+-2(2u_0)+O(1) since (6L)^2/h_*=2. Redistribute the principal real
linear tilt on that interval using
[
int_{-u_0}^{u_0}sgn(t)H(s(t)),dt
=-sum_{d':,|t_{d'}|<u_0}W_{d'}(u_0-|t_{d'}|),
]
-appearing with positive coefficient $12pi L/h_*$. Terms from purely
+appearing with positive coefficient 12pi L/h_*. Terms from purely
imaginary tangents have unit modulus for this leading tilt. All
-unfavorable sites have $W<0$ and hence the strong chemical suppression,
+unfavorable sites have W<0 and hence the strong chemical suppression,
which absorbs their loss at bounded additive cost per particle while
-retaining a net suppression $-c_1(u_0-|t|)$ per particle there
-($M_gge d$ and no species has $W=0$). We replace this real linear
+retaining a net suppression -c_1(u_0-|t|) per particle there
+(M_gge d and no species has W=0). We replace this real linear
tilt in the core edge weights by the point terms from the displayed
identity: they telescope on cycles. The remaining added terms have
-bounded-above point log moduli $1,2$, and at most bounded linear height
+bounded-above point log moduli 1,2, and at most bounded linear height
cost on cells whose real coefficients beyond the reference Gaussian
tend to zero away from the center after redistribution. There is no
winding boundary cost in this telescoping.
@@ -3016,15 +3000,15 @@ uniformly bounded after the scalar removal; we need the product
through the whole core to be bounded independently of its length. For
a core cell let the depth be its minimum distance from the two core
cuts, truncated below at zero. Reserve a fixed small fraction of the
-surviving negative point potential $-c_1(u_0-|t|)$: on cell $i$ put
-$p_i(x)=delta,depth_iN_x$ with fixed small $delta>0$, and
-$p_i=0$ on exterior states at the two core interfaces. If $K_i(x,y)$
+surviving negative point potential -c_1(u_0-|t|): on cell i put
+p_i(x)=delta,depth_iN_x with fixed small delta>0, and
+p_i=0 on exterior states at the two core interfaces. If K_i(x,y)
is the outgoing edge kernel, replace it by
[
K_i^{rm sym}(x,y)=exp!Bigl[frac{p_i(x)-p_{i+1}(y)}2Bigr]K_i(x,y).
]
The reserved exponent on such an edge is
-$-p_i(x)/2-p_{i+1}(y)/2$. In a product through the core the multipliers
+-p_i(x)/2-p_{i+1}(y)/2. In a product through the core the multipliers
telescope pointwise, including the interfaces where the exterior
factors are one, so the complete product kernel is unchanged and no
inverse diagonal gauge is required. Adjacent cell depths differ by at
@@ -3032,21 +3016,21 @@ most the fixed cell length, and the unused point suppression together
with the Gaussian margins of (eq:legacy-P-30) remains available
on both incident edges. Consequently every block with a nonempty
state at either endpoint has operator norm at most
-$Cexp(-c,depth)$, where the depth is the smaller of the two
+Cexp(-c,depth), where the depth is the smaller of the two
incident cell depths.
Empty-to-empty edges.: Away from a fixed number of cells
at the central transition and at the two cuts, the empty-to-empty
-block has norm at most $1$. Indeed, the integer height is constant
+block has norm at most 1. Indeed, the integer height is constant
across such an edge, the real reference exponent is
-$-pi^2S_0H^2/h_*$, and the residual linear coefficient of $H$ tends
+-pi^2S_0H^2/h_*, and the residual linear coefficient of H tends
exponentially to zero away from the transition and the cuts. Since
-$H$ is an integer, its absolute contribution is dominated by the
-negative quadratic whenever $Hne0$, and vanishes when $H=0$. The
+H is an integer, its absolute contribution is dominated by the
+negative quadratic whenever Hne0, and vanishes when H=0. The
common tangent is flat; remaining scalar phases have modulus one; the
state gauge cancels; and free Fock propagation is contractive. Thus
every complete edge has norm at most
-$1+Cexp(-c,depth)$ on each of the two core halves outside
+1+Cexp(-c,depth) on each of the two core halves outside
the fixed exceptional layers. For each depth there are only a bounded
number of cells, so the sum of these errors is bounded; taking the
product proves a uniform bound on the core operator. The exceptional
@@ -3058,22 +3042,22 @@ from its end cuts and the period edges,
(eq:legacy-P-32), (eq:legacy-P-33) and
(eq:legacy-P-25) converge in the edge-kernel sense, uniformly,
to the homogeneous kernels with (eq:legacy-P-29) and phases
-(eq:legacy-P-27): the undesired $B$-species are suppressed, and
-$A$-points see the chemical limit $1$. The direct left factors with
+(eq:legacy-P-27): the undesired B-species are suppressed, and
+A-points see the chemical limit 1. The direct left factors with
the linear term in (eq:legacy-P-33) give exactly
-$exp(2a_pm c_i)$; the pair kernels are unchanged on the interior
+exp(2a_pm c_i); the pair kernels are unchanged on the interior
flat cells (their common displacement cancels). Convergence in
Hilbert--Schmidt norm as distances increase follows from bounded-state
-truncation and the Gaussian majorants. Use the block length $q$
+truncation and the Gaussian majorants. Use the block length q
chosen after Lemma~lem:u-homogeneous-operator, for which each
-homogeneous $q$-fold product has norm at most
-$exp((1/2-ε)qS_0)$. Increase the fixed boundary layers
-until the $q$-fold products of the actual tail edges differ in norm
+homogeneous q-fold product has norm at most
+exp((1/2-ε)qS_0). Increase the fixed boundary layers
+until the q-fold products of the actual tail edges differ in norm
from their homogeneous products by less than the available fixed
-margin; they are then at most $exp(qS_0/2)$. The boundary layers and
-the fewer than $q$ remaining edges have bounded length and bounded
+margin; they are then at most exp(qS_0/2). The boundary layers and
+the fewer than q remaining edges have bounded length and bounded
norms. The two tails together contribute at most
-$(R-2u_0)/2+O(1)$ to the logarithm of the product norm; short
+(R-2u_0)/2+O(1) to the logarithm of the product norm; short
intervals and all remaining boundary layers use the uniform edge
bounds directly. In the cyclic trace estimate keep two edges in
Hilbert--Schmidt norm to use the remaining product bounds. Including
@@ -3090,7 +3074,7 @@ Matching the scalar and the Pfaffian normalization.:
The interval bound controls the particle integral. To recover the
polygon observable we must still match its scalar factors with the
-normalizing polynomial $f_N^2$, even when the staircase uses only two
+normalizing polynomial f_N^2, even when the staircase uses only two
rapidity values, because those values may have arbitrarily unequal
multiplicities.
@@ -3099,76 +3083,76 @@ External scalars.: Uniformly on the circle,
[
tfrac12sum_{e,f}I_{g_eg_f}(v_e-v_f)+C_2e^{-2T}+O(1)
]
-for a site-data constant $C_2$, where
+for a site-data constant C_2, where
[
I_{gh}(v)=2q_gq_hlog2+int_{mathbb R}bbR!!
[K_{gh}(p)widehatrho(p)^2cosh(vp)-4q_gq_h/p^2],(dp)/(2pi).
]
-The subtracted pole all-mode sum is $q_gq_hT/3$. Contour shifting
-with exponentially bounded tails gives error $O(N^2e^{-a_0Re T})$,
-picking corrections at poles $pm2i$, and at most a simple pole at
-$pm ia_0$ in the color-difference channel. Then
+The subtracted pole all-mode sum is q_gq_hT/3. Contour shifting
+with exponentially bounded tails gives error O(N^2e^{-a_0Re T}),
+picking corrections at poles pm2i, and at most a simple pole at
+pm ia_0 in the color-difference channel. Then
e^{I_{gh}(v)}=B(v)^2tan^2(lambda_bv)
1/D(v),&g=h,1,&gne h,
-quad B(v)=(sin(v+b)sin(v-b))/(sin v).
+ B(v)=(sin(v+b)sin(v-b))/(sin v).
-Verification: for real $v$ the multiplier
-$(p/2pi)K_{gh}widehatrho^2$ on $p>0$ is
-$t^2/[(1-t+t^2)(1-t^8)]$ times $1+t^6$ (same color) or $-2t^3$
-(opposite), with $t=e^{-Lp}$. Thus it expands as
-$sum_{kge1}c_ke^{-kLp}$ with
+Verification: for real v the multiplier
+(p/2pi)K_{gh}widehatrho^2 on p>0 is
+t^2/[(1-t+t^2)(1-t^8)] times 1+t^6 (same color) or -2t^3
+(opposite), with t=e^{-Lp}. Thus it expands as
+sum_{kge1}c_ke^{-kLp} with
[
c_k=-2{bf1}_{k=pm3(8)}+2{bf1}_{k=0(8)}
+2{bf1}_{k=3(6)}-2{bf1}_{k=0(6)}
+{bf1}_{g=h}{bf1}_{k=pm2,pm3(8)}.
]
-Integrating gives $sum_k-c_klog(1-v^2/(kL)^2)$, matching
+Integrating gives sum_k-c_klog(1-v^2/(kL)^2), matching
(eq:legacy-P-35) by the sine product; the constant follows by
-$v=iV$, $Vto+∞$, using $I_{gh}(iV)=2q_gq_h(log2-V)+o(1)$.
+v=iV, Vto+∞, using I_{gh}(iV)=2q_gq_h(log2-V)+o(1).
The magnetic test scalar.: Formula (eq:legacy-P-31)
reduces to
[
sum_elog(J(x_e-v_e))/(J(v_e-x_e))+C_1e^{-T}+O(1).
]
-The regular test variance costs $O(1)$. The cross part integrates $a$
+The regular test variance costs O(1). The cross part integrates a
against the shifted periodized densities; the log of the normalized
-squared edge ratio gives first terms $e^{-T}(d_1e^{2s}+d_2e^{-2s})$
-across the core, with errors $O(e^{-2Re T+4|Re s|})$, whose
-convolution with each $rho(s+iv_e)$ contributes the coefficient and
-costs $O(1+Ne^{-a_0Re T/2})$. The principal per-background integral
-is the whole-line core-log convolution shifted by $i(x_e-v_e)$. Its
+squared edge ratio gives first terms e^{-T}(d_1e^{2s}+d_2e^{-2s})
+across the core, with errors O(e^{-2Re T+4|Re s|}), whose
+convolution with each rho(s+iv_e) contributes the coefficient and
+costs O(1+Ne^{-a_0Re T/2}). The principal per-background integral
+is the whole-line core-log convolution shifted by i(x_e-v_e). Its
derivative has transform
[
2pi iwidehatrho(p)(z-z^7)/(1-z^8)
-=2pi i(z^2+z^3-z^5-z^6)/(1-z^8),quad z=e^{-Lp},
+=2pi i(z^2+z^3-z^5-z^6)/(1-z^8), z=e^{-Lp},
]
obtained from the localized coth difference; hence the convolution is
-$log[J(-is)/J(is)]$ at $s=i(x_e-v_e)$ (matching at $s=0$ by oddness,
+log[J(-is)/J(is)] at s=i(x_e-v_e) (matching at s=0 by oddness,
and shifting contours within the strip keeps the two pole sets
separated).
-Determinant scalar.: Remove $C_2e^{-2T}+C_1e^{-T}$ by an
+Determinant scalar.: Remove C_2e^{-2T}+C_1e^{-T} by an
exponential analytic multiplier and use maximum modulus. The
-two-color transform determinant adds $(3/8)log N+O(1)$. For the
-remaining determinant put $mathfrak d(p)=-frakd(p)=-log(1-e(p))$ and use
+two-color transform determinant adds (3/8)log N+O(1). For the
+remaining determinant put mathfrak d(p)=-frakd(p)=-log(1-e(p)) and use
[
2sum_{nge1}mathfrak d(2frakd(2pi n/T)=sum_{ninmathbb Z}bbZ
mathfrak d(2frakd(2pi n/T)-mathfrak d(0).frakd(0).
]
-Since $mathfrak d$frakd$ is even, regular at zero, and exponentially
+Since mathfrak dfrakd is even, regular at zero, and exponentially
decaying, the periodization contributes
-$pi T/(16L)-mathfrak d(0)+O(e^frakd(0)+O(e^{-cRe T})$ as in the two-line case.
+pi T/(16L)-mathfrak d(0)+O(e^frakd(0)+O(e^{-cRe T}) as in the two-line case.
The two square-root factors have constant product. With
-$Re T=(3/4)log N+O(1)$ this gives the stated uniform bound on the
+Re T=(3/4)log N+O(1) this gives the stated uniform bound on the
nome circle. Formula (eq:legacy-P-35), with
-$e^{I_s(0)/2}=lambda_b/sqrt C$, cancels the all-positive prefactors
+e^{I_s(0)/2}=lambda_b/sqrt C, cancels the all-positive prefactors
in (eq:legacy-P-23) along with the site sum from
-(eq:legacy-P-31), leaving exactly $2f_N^{(0)2}$ in the numerator
-relative to $2f_N^2$, where $f_N^{(0)}$ uses
-$h_0(v)=-lambda_btan(lambda_bv)/2$ instead of $h$ (also by
+(eq:legacy-P-31), leaving exactly 2f_N^{(0)2} in the numerator
+relative to 2f_N^2, where f_N^{(0)} uses
+h_0(v)=-lambda_btan(lambda_bv)/2 instead of h (also by
Schur--Pfaffian at shifted variables).
The Pfaffian ratio.: We need
@@ -3177,26 +3161,26 @@ log(f_N^2/f_N^{(0)2})ge(5/24)log N-O(1).
-Shifted comparison. Center the variables in $[-L/2,L/2]$ and
+Shifted comparison. Center the variables in [-L/2,L/2] and
make them distinct (or perturb into a slightly larger interval). The
kernels, by their partial fractions, are integrations of
-$-sinh(vp)$ against the positive-ray pole densities $nu_0,nu$ of
-ratio $R(p)=nu/nu_0ge1$. Expanding a Pfaffian of size $N=2s$ by
+-sinh(vp) against the positive-ray pole densities nu_0,nu of
+ratio R(p)=nu/nu_0ge1. Expanding a Pfaffian of size N=2s by
antisymmetry gives, up to common constants, the integral of the
-alternant on exponents $pm p_1,...,pm p_s$ against the
+alternant on exponents pm p_1,...,pm p_s against the
corresponding product measure. With exponents laid out in pairs the
alternant has fixed sign for distinct ordered variables, since
-exponential polynomials of $N$ such distinct exponents have at most
-$N-1$ real zeros by Rolle induction. Jensen then bounds the log
-Pfaffian ratio below by its linear statistic with $g=log R$ in the
-$h_0$ ensemble, equivalently the log derivative of
-$Pf(h_0+th_g)$ at zero, where $h_g$ uses pole density
-$gnu_0$; confluent limits preserve the comparison.
-
-Uniform $Alog N+O(1)$. The doubled linear statistic has the
-form $Alog N+O(1)$, with $A$ independent of the rapidities and of
-the even size $N$. Put $H_t=[h_0(v_i-v_j)+th_g(v_i-v_j)]_{i,j=1}^N$
-and $C(v)=prod_j[-icot(lambda_b(v-v_j))]$. If $Gamma$ positively
+exponential polynomials of N such distinct exponents have at most
+N-1 real zeros by Rolle induction. Jensen then bounds the log
+Pfaffian ratio below by its linear statistic with g=log R in the
+h_0 ensemble, equivalently the log derivative of
+Pf(h_0+th_g) at zero, where h_g uses pole density
+gnu_0; confluent limits preserve the comparison.
+
+Uniform Alog N+O(1). The doubled linear statistic has the
+form Alog N+O(1), with A independent of the rapidities and of
+the even size N. Put H_t=[h_0(v_i-v_j)+th_g(v_i-v_j)]_{i,j=1}^N
+and C(v)=prod_j[-icot(lambda_b(v-v_j))]. If Gamma positively
encircles the sites and initially no other poles,
[
2Bigl.(d)/(dt)log(Pf H_t)/(Pf H_0)|_{t=0}
@@ -3204,20 +3188,20 @@ encircles the sites and initially no other poles,
C(v)C(w)h_0(v-w)h_g(v-w),dv,dw.
]
Cancelling common Schur-minor factors gives the undoubled derivative
-$-4sum_{i<j}r_ir_jh_0(v_i-v_j)h_g(v_i-v_j)$ with
-$r_i=Res_{v=v_i}C$; the normalized double integral is
-$2sum r_ir_jh_0h_g$ because diagonal residues vanish (both kernels
-odd). Doubling thus gives $-4$ independently of sites and even $N$.
-
-The odd $h_g$ is bounded analytic on strict substrips $|Re v|<5L$ by
-exponential decay of $gnu_0$; its $1/p$ portion has bounded odd sine
-integral with cutoff. Expand the $v$ contour to a rectangle with
-vertical edges at $pm2.2L$ and heights $pm a_0^{-1}log N$; do $w$
+-4sum_{i<j}r_ir_jh_0(v_i-v_j)h_g(v_i-v_j) with
+r_i=Res_{v=v_i}C; the normalized double integral is
+2sum r_ir_jh_0h_g because diagonal residues vanish (both kernels
+odd). Doubling thus gives -4 independently of sites and even N.
+
+The odd h_g is bounded analytic on strict substrips |Re v|<5L by
+exponential decay of gnu_0; its 1/p portion has bounded odd sine
+integral with cutoff. Expand the v contour to a rectangle with
+vertical edges at pm2.2L and heights pm a_0^{-1}log N; do w
similarly with imaginary half-height one larger. The only crossed
-poles are from $h_0$ at $v-w=pm b$, with $C(v)C(vpm b)=1$, so their
+poles are from h_0 at v-w=pm b, with C(v)C(vpm b)=1, so their
contributions are site-independent. The remaining integral is
-uniformly bounded: $|C|$ on vertical edges is at most
-$exp(-cNe^{-a_0|Im v|})$, horizontal caps are bounded, and the only
+uniformly bounded: |C| on vertical edges is at most
+exp(-cNe^{-a_0|Im v|}), horizontal caps are bounded, and the only
simple poles on the contour product are transversal edge-cap
intersections (locally absolutely integrable). The growing vertical
sides cost
@@ -3225,16 +3209,16 @@ sides cost
int_0^{a_0^{-1}log N}e^{-cNe^{-a_0t}},dt
=a_0^{-1}int_1^N e^{-cx}(dx)/(x)=O(1),
]
-and the extra $w$-height unit adds only a bounded factor. Each
-crossed pole $w=vpm b$ contributes a constant residue. This proves
-$Alog N+O(1)$ uniformly in the rapidities, including confluent
+and the extra w-height unit adds only a bounded factor. Each
+crossed pole w=vpm b contributes a constant residue. This proves
+Alog N+O(1) uniformly in the rapidities, including confluent
limits (after the usual division by vanishing Vandermonde factors).
-Coefficient $Age5/24$. At coincidence, twice the linear
+Coefficient Age5/24. At coincidence, twice the linear
statistic is exactly the compressed trace test from the homogeneous
saturation argument (Taylor row and column differences). That test
-has lower limiting coefficient $5/24$, obtained there from Fatou and
-the explicit orthogonal polynomial kernel. Hence $Age5/24$, and
+has lower limiting coefficient 5/24, obtained there from Fatou and
+the explicit orthogonal polynomial kernel. Hence Age5/24, and
applying Jensen at the shifted sites proves (eq:legacy-P-36)
with a bounded additive error.
@@ -3243,19 +3227,19 @@ is at most
[
-2R+tfrac52(R-2u_0)+(frac38-frac5{24})log N+O(1).
]
-Choosing $R=(3/4)log N+O(1)$ and $2u_0=(3/4)log d+O(1)$ makes this
+Choosing R=(3/4)log N+O(1) and 2u_0=(3/4)log d+O(1) makes this
[
-tfrac43log N+(15)/(8)log(N/d)+O(1).
]
Exponentiation proves the uniform two-edge bound
(eq:marked-uniform). If the shorter core has only a bounded
number of cells, use the uniform edge bounds there; enlarging the
-absolute constant covers small $d$.
+absolute constant covers small d.
Exterior sewing and finite bridge moments.:
-Everything here uses pure sides (lattice cuts normal to the $n_j$),
+Everything here uses pure sides (lattice cuts normal to the n_j),
and lengths count visited triangle centres. The marked-polygon bound
controls two-link events; we first construct exterior connectors, then
recover their costs by averaging the cuts, and use them to turn the
@@ -3264,18 +3248,17 @@ two-mark estimate into a second length moment.
[Confined crossings and half-plane monotonicity]
The following statements hold with constants depending only on the
fixed relative tolerances and pure normals.
-
-item Starting at a mid-edge source with inward normal $n$, one can
-travel in that half-plane to a pure terminal line at distance $s$ whose
-outward normal $m$ makes angle at most $60^circ$ with $n$, staying
+ Starting at a mid-edge source with inward normal n, one can
+travel in that half-plane to a pure terminal line at distance s whose
+outward normal m makes angle at most 60^circ with n, staying
before the terminal line until the end. For any fixed relative
-tolerance and all sufficiently large admissible $s$, the travel is
+tolerance and all sufficiently large admissible s, the travel is
confined within that tolerance around the straight segment in
-direction $m$, with mass $gtrsim s^{-1/4}$; the unrestricted mass in
-this two-cut domain is $O(s^{-1/4})$.
-item From one mid-edge source to another on the same pure boundary at
-distance $l>0$, half-plane arches of diameter $O(l)$ have mass
-$gtrsim l^{-5/4}$; even without confinement the mass is $O(l^{-5/4})$.
+direction m, with mass gtrsim s^{-1/4}; the unrestricted mass in
+this two-cut domain is O(s^{-1/4}).
+ From one mid-edge source to another on the same pure boundary at
+distance l>0, half-plane arches of diameter O(l) have mass
+gtrsim l^{-5/4}; even without confinement the mass is O(l^{-5/4}).
The boundary-to-boundary mass is decreasing as the terminal port moves
away from the source along one straight side of a convex pure-sided
@@ -3283,30 +3266,30 @@ truncation, whenever the compared adjacent ports remain on that side.
Proof.
For 1, the upper bound is the convex diameter bound. For the lower
-bound with a turn, choose small fixed $a>0$ in terms of the confinement
-tolerance. Exit first through the side normal to $m$ in the centered
-triangle of size $s_1$, averaged over $s_1in[as,2as]$; the deep-exit
-estimate (eq:deep-triangle-exit) retains mass $gtrsim s^{-1/4}$
-at depth $ge$ a smaller fixed multiple of $s$. Then bridge along $m$
+bound with a turn, choose small fixed a>0 in terms of the confinement
+tolerance. Exit first through the side normal to m in the centered
+triangle of size s_1, averaged over s_1in[as,2as]; the deep-exit
+estimate (eq:deep-triangle-exit) retains mass gtrsim s^{-1/4}
+at depth ge a smaller fixed multiple of s. Then bridge along m
to the terminal, confined within a tolerance much smaller than this
initial depth (so the confinement stays inside the initial half-plane),
-costing $gtrsim s^{-1/4}$ more. Joint mass for two choices of the
+costing gtrsim s^{-1/4} more. Joint mass for two choices of the
first level splits at both transversal renewals into the triangle
exit, the difference bridge, and the final bridge, averaging therefore
-$O(s^{-3/4})$; Cauchy--Schwarz gives mass
-$gtrsim (s^{-1/2})^2/s^{-3/4}=s^{-1/4}$. Without a turn use the
+O(s^{-3/4}); Cauchy--Schwarz gives mass
+gtrsim (s^{-1/2})^2/s^{-3/4}=s^{-1/4}. Without a turn use the
narrow-channel bound.
For 2, monotonicity via an artificial arc in a convex pure-sided
-truncation: from $a<P<Q$ on the same straight side ($P,Q$ adjacent),
-the arc input from $ato Q$ back towards $P$ can only exit between
-$a,P$, with phase $z^{-1}$; conversely from $ato P$ via the arc to
-$Q$, the residual chord has $|W|lepi$. After multiplication by
-$z=e^{ibetapi}$ the outgoing imaginary coefficients are therefore
+truncation: from a<P<Q on the same straight side (P,Q adjacent),
+the arc input from ato Q back towards P can only exit between
+a,P, with phase z^{-1}; conversely from ato P via the arc to
+Q, the residual chord has |W|lepi. After multiplication by
+z=e^{ibetapi} the outgoing imaginary coefficients are therefore
nonnegative as in the earlier adjacent-cap argument. In a large
-centered half-hexagon of scale $asymp l$, the loss from full
-half-plane arches to ends at distances in $[l,2l]$ is small compared
-with their total mass of order $l^{-1/4}$ (by the diameter bound).
+centered half-hexagon of scale asymp l, the loss from full
+half-plane arches to ends at distances in [l,2l] is small compared
+with their total mass of order l^{-1/4} (by the diameter bound).
Apply monotonicity inside the truncation.
@@ -3314,57 +3297,56 @@ Two exterior sewing estimates.:
Take a triangular-lattice vertex as origin and pure-support domains
-$D=D(0;mathbf b)$bfb)$ with $|b_j-R|leε R$ for a fixed
-$0<εle1/20$; these are convex hexagons with minimum side
-$asymp R$, and Section~sec:hexagon-completion supplies uniform
+D=D(0;mathbf b)bfb) with |b_j-R|leε R for a fixed
+0<εle1/20; these are convex hexagons with minimum side
+asymp R, and Section~sec:hexagon-completion supplies uniform
nesting and central-occupancy estimates for every such domain,
including independent bounded row roundings.
-Use a random nested family $D_h$ indexed by $h$ in a window of size
-$cR$, $c>0$ small. Move each nominal support outward at speed $1$
-from a regular arrangement plus a perturbation $c'Rf(V_i+A_ih/R)$,
-with $c'>0$ small, $f$ a unit-period triangular zigzag of slope $1$,
-and $V_i$ uniform, $A_i$ uniform on $[-1,1]$, independently. Given
-$r<s$ in that window, conditional on the phases at $r$, the nominal
-supports at $s$ are independent with densities at most $C/(s-r)$.
+Use a random nested family D_h indexed by h in a window of size
+cR, c>0 small. Move each nominal support outward at speed 1
+from a regular arrangement plus a perturbation c'Rf(V_i+A_ih/R),
+with c'>0 small, f a unit-period triangular zigzag of slope 1,
+and V_i uniform, A_i uniform on [-1,1], independently. Given
+r<s in that window, conditional on the phases at r, the nominal
+supports at s are independent with densities at most C/(s-r).
For each facet choose a fixed nondecreasing rounding map to its
admissible row levels with uniformly bounded error. Each nominal
-support has derivative at least $1-c'>0$, so rounding preserves
+support has derivative at least 1-c'>0, so rounding preserves
nesting and preserves independence between facets conditional on inner
-phases. If its error is at most $Kd_0$, an atom of the rounded
-support is the image of an interval of length at most $2Kd_0$; hence
+phases. If its error is at most Kd_0, an atom of the rounded
+support is the image of an interval of length at most 2Kd_0; hence
the conditional density bound becomes an atom bound
-$Cmin(1,Kd_0/(s-r))$, sufficient for all averages below. Boundary
-gaps are $O(1+s-r)$ and at least $c_0(s-r)-O(1)$.
-
-Use complete chords in $D_h$ with ordered boundary ends $p,q$ binned
-by dyadic separation $lasymp|p-q|$ (at least constant order) and by
-the two values $t_i$ of $1+$ distance to the nearest corner capped at
-order $l$, within dyadic factors. Orientations do not affect any
-bound. Fix a chord event of mass $E_h$ in such a bin; its contents
+Cmin(1,Kd_0/(s-r)), sufficient for all averages below. Boundary
+gaps are O(1+s-r) and at least c_0(s-r)-O(1).
+
+Use complete chords in D_h with ordered boundary ends p,q binned
+by dyadic separation lasymp|p-q| (at least constant order) and by
+the two values t_i of 1+ distance to the nearest corner capped at
+order l, within dyadic factors. Orientations do not affect any
+bound. Fix a chord event of mass E_h in such a bin; its contents
(such as marked vertices) may be constrained arbitrarily as a
-deterministic event given $h,D_h$.
+deterministic event given h,D_h.
[Exterior sewing]
For the nested family, endpoint bin, and deterministic chord event
-just specified, the average $overlineoverline E$E$ over the level and support
+just specified, the average overlineoverline EE over the level and support
randomness satisfies, with constants uniform in the event and the
admissible levels:
-
-item If it requires the use of two fixed internal links, then
-$overlineoverline EEle Cl^{3/4}mathcal K$,calK$, where $mathcal K$calK$ is the
+ If it requires the use of two fixed internal links, then
+overlineoverline EEle Cl^{3/4}mathcal K,calK, where mathcal KcalK is the
single-polygon mass using those links on a cylinder of period vector
in any lattice direction chosen for the marks and length large
-compared with $R$.
-item For $l$ large, suppose the goal is instead to extend to bridges
-between horizontal caps at heights near $pm AR$, $A$ a fixed large
-constant (each height magnitude in $[AR,2AR]$). Let $B^*$ count
+compared with R.
+ For l large, suppose the goal is instead to extend to bridges
+between horizontal caps at heights near pm AR, A a fixed large
+constant (each height magnitude in [AR,2AR]). Let B^* count
bridges between those caps, with base start free in a horizontal
-interval of size $CR$ containing the required chord event for at
+interval of size CR containing the required chord event for at
least one choice of the supports. Then
-$B^*ge cR^{-1/2}(l/R)^{C_0}l^{1/2}overlineoverline E$,E$, for universal
-$c,C_0>0$.
+B^*ge cR^{-1/2}(l/R)^{C_0}l^{1/2}overlineoverline E,E, for universal
+c,C_0>0.
Proof.
@@ -3372,26 +3354,26 @@ The two completions are shown in Figure~fig:sewing.
input{figures/sewing}
Exterior connectors. For 1, use a connector strictly in the
-exterior with diameter at most $Cl$, fixed endpoints $p,q$, mass
-available $ge cl^{-5/4}$. For 2, append from $p,q$ separate
+exterior with diameter at most Cl, fixed endpoints p,q, mass
+available ge cl^{-5/4}. For 2, append from p,q separate
exterior paths to the two caps without touching either earlier, also
-within distance $CR$ of the origin; together their available mass is
-at least $cR^{-1/2}exp[-C(1+log^+(R/l))]$. In both cases, for
-large $l$, arrange both exterior branches to advance from their
-endpoints in disjoint small patches until reaching distance $rho l$
-from $D_h$, and all exterior portions between or beyond these visits
-stay at distance $gerho l$ ($rho>0$ fixed small). Each enlarged
+within distance CR of the origin; together their available mass is
+at least cR^{-1/2}exp[-C(1+log^+(R/l))]. In both cases, for
+large l, arrange both exterior branches to advance from their
+endpoints in disjoint small patches until reaching distance rho l
+from D_h, and all exterior portions between or beyond these visits
+stay at distance gerho l (rho>0 fixed small). Each enlarged
patch sees only facets of a single consecutive-normal pair for the
outer and inner cuts used below.
-For 1, work in units $l$. Initial straight pieces along outward pure
+For 1, work in units l. Initial straight pieces along outward pure
normals, then two disjoint simple routes in the open exterior
approaching a common terminal pure line from the same side in the
pure perpendicular direction at close separated points; around the
terminal patch keep a free half-ball of radius much greater than the
terminal separation (so the preceding confined-arch lemma applies).
Other pieces are straight in pure directions with successive turns
-$le60^circ$, each travelled by the point-to-line lemma with small
+le60^circ, each travelled by the point-to-line lemma with small
tolerances. By compactness of subsequences under scaling and
translation, initial rays give strictly positive distance by their
supporting lines; one then follows disjoint smooth arcs in the open
@@ -3407,39 +3389,39 @@ successive crossing neighbourhoods are separated, and adjacent pieces
stay in their required half-planes at each transverse crossing. This
works along the subsequence, including the localized initial deep
excursions, and gives uniform lengths, step counts, and apertures at
-this scale. For bounded $l$, a bounded-cost local join outside
+this scale. For bounded l, a bounded-cost local join outside
suffices.
-For 2 with $l$ a fixed small multiple of $R$ or larger, the same
-construction in units $R$ without a terminal arch follows two disjoint
-paths to the two caps, terminating separately. For smaller $l$, the
+For 2 with l a fixed small multiple of R or larger, the same
+construction in units R without a terminal arch follows two disjoint
+paths to the two caps, terminating separately. For smaller l, the
points lie on the same or adjacent facets: start along divergent pure
-directions (on the same facet, rotate $m$ by $60^circ$ in opposite
+directions (on the same facet, rotate m by 60^circ in opposite
directions; on adjacent facets use their two outward normals).
-Distances along the chosen straight rays at parameters $x,yge0$ are
-separated by $ge c(l+x+y)$ by projection, and supporting-normal
-height increases at rate $ge1/2$. Travel to a small fixed distance
-of order $R$ along each with a point-to-line crossing at scale $l$,
+Distances along the chosen straight rays at parameters x,yge0 are
+separated by ge c(l+x+y) by projection, and supporting-normal
+height increases at rate ge1/2. Travel to a small fixed distance
+of order R along each with a point-to-line crossing at scale l,
then confined pure bridges along the ray to successively doubled
heights; error accumulation is geometric, so small tolerances work
simultaneously. The continuation from macroscopically separated
-locations by finitely many $R$-scale legs follows as above;
+locations by finitely many R-scale legs follows as above;
uniformity holds even if the rays had coincident limiting starts,
since their terminal segments are outgoing and separated.
Averaging intermediate levels. Fix one corridor construction
-with $D_h,p,q$ fixed. In both cases, vary each intermediate
+with D_h,p,q fixed. In both cases, vary each intermediate
transverse level uniformly over admissible levels, in its own small
independent window of order the local scale, the terminal pure line
-for case 1 shared at both ends. Let $nu$ be critical weight on all
+for case 1 shared at both ends. Let nu be critical weight on all
exterior objects with the stated confinement and depth (paths
-$pto q$ in case 1, or ordered pairs of disjoint paths to the two
+pto q in case 1, or ordered pairs of disjoint paths to the two
assigned caps in case 2), counting all centres in exterior paths.
-For a cut list $mathbf u$,bfu$, let
-$mathbf1_bf1_{mathbf u}bfu(eta)$ indicate that $eta$ follows the
+For a cut list mathbf u,bfu, let
+mathbf1_bf1_{mathbf u}bfu(eta) indicate that eta follows the
prescribed corridors at those levels, and put
-$Y(eta)=mathbb E_bbE_{mathbf u}bfumathbf1_bf1_{mathbf u}bfu(eta)$ and
-$P=prod_{j=1}^ks_j^{-1/4}$. Concatenating the confined crossings
+Y(eta)=mathbb E_bbE_{mathbf u}bfumathbf1_bf1_{mathbf u}bfu(eta) and
+P=prod_{j=1}^ks_j^{-1/4}. Concatenating the confined crossings
gives
[
int Y,dnuge
@@ -3454,46 +3436,46 @@ from one end, the two incident half-plane constraints put all
crossings of that stage in order, with an intervening portion
travelling to the next stage between its respective lines (a distance
comparable to its scale). The intervening costs with incoming point
-prescribed are $O(s_j^{-1/4})$ by the upper point-to-line estimate.
+prescribed are O(s_j^{-1/4}) by the upper point-to-line estimate.
In case 1 the completing arch at the later shared level costs
-$O(l^{-5/4})$ for any attained endpoints; averaging the joint costs
-supplies an extra $s_j^{-1/4}$ for each varied renewal per branch at
-scale $s_j$, which stays integrable. In case 2 only the two final
+O(l^{-5/4}) for any attained endpoints; averaging the joint costs
+supplies an extra s_j^{-1/4} for each varied renewal per branch at
+scale s_j, which stays integrable. In case 2 only the two final
cap levels are fixed, their endpoints free, and both final legs of
-scale $asymp R$. Consequently
+scale asymp R. Consequently
[
int Y^2,dnule
C^kl^{-5/4}P^2,&case 1,
C^kR^{1/2}P^2,&case 2,
]
-the factor $R^{1/2}$ in case 2 having removed from the second copy of
-$P$ the two factors at the fixed cap levels. Cauchy--Schwarz gives
+the factor R^{1/2} in case 2 having removed from the second copy of
+P the two factors at the fixed cap levels. Cauchy--Schwarz gives
[
nu{Y>0}ge((int Y,dnu)^2)/(int Y^2,dnu)ge
c_1^kl^{-5/4},&case 1,
c_1^kR^{-1/2},&case 2,
]
-for a fixed $0<c_1<1$. The number of legs is bounded in case 1, and
-in case 2 at most $C(1+log^+(R/l))$, so
-$c_1^kge cmin(1,(l/R)^{C_0})$ for a fixed $C_0>0$. These are the
+for a fixed 0<c_1<1. The number of legs is bounded in case 1, and
+in case 2 at most C(1+log^+(R/l)), so
+c_1^kge cmin(1,(l/R)^{C_0}) for a fixed C_0>0. These are the
asserted connector masses.
Common target measure. Trim the exterior choices by fractional
-indicators to a common lower mass $Fasymp l^{-5/4}$ (case 1) or
-$Fasymp R^{-1/2}min(1,(l/R)^{C_0})$ (case 2), pointwise for
-$p,q,h,D_h$. The augmented object is a polygon injecting into the
+indicators to a common lower mass Fasymp l^{-5/4} (case 1) or
+Fasymp R^{-1/2}min(1,(l/R)^{C_0}) (case 2), pointwise for
+p,q,h,D_h. The augmented object is a polygon injecting into the
cylinder in case 1, or a full bridge in case 2; a fixed cut choice
recovers the decomposition up to bounded multiplicity since there are
-exactly two boundary crossings. Let $Ω$ be the six
-phase--slope pairs with the product law and $mu$ critical weight on
+exactly two boundary crossings. Let Ω be the six
+phase--slope pairs with the product law and mu critical weight on
target polygons through the prescribed links (case 1) or target
bridges satisfying the common deterministic event (case 2), with
-total mass $lemathcal K$calK$ or $B^*$. In a level window $I$ of size
-$asympε l$, let $b_h(Gamma,Ω)$ sum the recovered
-ordered decompositions of $Gamma$ at $∂ D_h$ of the
+total mass lemathcal KcalK or B^*. In a level window I of size
+asympε l, let b_h(Gamma,Ω) sum the recovered
+ordered decompositions of Gamma at ∂ D_h of the
chord-event indicator times the exterior trimming factor, so
-$0le b_hle C$; put
-$X(Gamma,Ω)=average_{hin I}b_h(Gamma,Ω)$.
+0le b_hle C; put
+X(Gamma,Ω)=average_{hin I}b_h(Gamma,Ω).
Then
int X,d(mu ⊗ mathbb P_Ω)bbP_Ω)asymp Foverlineoverline E,E,
@@ -3502,18 +3484,18 @@ int X,d(mu ⊗ mathbb P_Ω)bbP_Ω)asymp Foverlineoverline E,E,
where the left side includes the level average and support
randomness.
-Choose two levels independently and uniformly in $I$; by symmetry
-integrate over $r<s$, conditioned on the product measure restricted
+Choose two levels independently and uniformly in I; by symmetry
+integrate over r<s, conditioned on the product measure restricted
to that half-square, plus any diagonal atoms. This restriction is not
-renormalized: conditional on $r$, the remaining $s$-measure (or its
-rounded atom mass) has density at most $C/l$. For $ε>0$
+renormalized: conditional on r, the remaining s-measure (or its
+rounded atom mass) has density at most C/l. For ε>0
sufficiently small, a simultaneous event splits into the chord event
-at $D_r$, two exterior bridges crossing the band to $∂ D_s$,
+at D_r, two exterior bridges crossing the band to ∂ D_s,
and the augmentation from there; keep the trimming factor from the
latter. Boundary crossings are unique on the separate branches,
localized by the depth condition since the boundary moves by at most
-$Cε l+O(1)$. For bounded $s-r$ or bounded $l$, the indicator
-at $r$ alone suffices. Bounding the averaged cost $Xi$ of the two
+Cε l+O(1). For bounded s-r or bounded l, the indicator
+at r alone suffices. Bounding the averaged cost Xi of the two
band bridges uniformly conditional on the inner level and phases
gives
@@ -3521,12 +3503,12 @@ int X^2,d(mu ⊗ mathbb P_Ω)bbP_Ω)le CFoverlineoverline EE,Xi.
-The two band bridges. Put $g=s-r$. Fix for each bridge the
+The two band bridges. Put g=s-r. Fix for each bridge the
possible target normal in the local facet pair. Drop all other outer
cuts, keeping the inner elbow in a local patch with two lateral pure
trims transversal to both tangent directions containing the bridge
portions; the domain is a polygonal strip between this outward cut
-line and the flat or elbow piece of $D_r$. The inner obstacle keeps
+line and the flat or elbow piece of D_r. The inner obstacle keeps
just the two corresponding consecutive constraints, extended through
the strip patch. The target outward cut line is disjoint from it by
the positive gap; the trims may for instance bound projection on the
@@ -3534,50 +3516,50 @@ common tangent bisector, wide enough to enclose the localized portions
and leave clearance at the source.
There is a positive contour (after a possible rotation) from the inner
-source. Closing a chord along the boundary gives $W$ equal to lifted
-outward-normal change minus $pi$ around the perimeter from source to
+source. Closing a chord along the boundary gives W equal to lifted
+outward-normal change minus pi around the perimeter from source to
exit. There is only one reflex corner on the inner elbow (outward
-normal decreasing by $pi/3$), and its two outward normals and the
+normal decreasing by pi/3), and its two outward normals and the
outward normals on the trim sides and target line are in cyclic order
apart from that corner (the trims use a pure direction strictly
between opposing boundary sectors). Hence from a given inner source
-at most one sign of excess beyond $[-pi,pi]$ occurs, and by at most
-$pi/3$; rotating gives real coefficients bounded below positively,
-since $beta=3/8$. Convexity is not needed for the contour identity
+at most one sign of excess beyond [-pi,pi] occurs, and by at most
+pi/3; rotating gives real coefficients bounded below positively,
+since beta=3/8. Convexity is not needed for the contour identity
on this simply connected domain. Cutting down by a centered triangle
-at scale a small multiple of $min(g,t_i)$ (inner source on a flat
+at scale a small multiple of min(g,t_i) (inner source on a flat
side) and using contour subtraction and the escape upper estimate, the
-bridge cost is $O((1+min(g,t_i))^{-1/4})$, also by the full positive
+bridge cost is O((1+min(g,t_i))^{-1/4}), also by the full positive
contour when the minimum is bounded.
-Corner gain. Put $t=min_it_i$. Both tests in $X^2$ use the
-same dyadic endpoint bins: the two inner bins have sizes at least $t$,
-and one outer bin has size of order $t$. Sum over the finitely many
+Corner gain. Put t=min_it_i. Both tests in X^2 use the
+same dyadic endpoint bins: the two inner bins have sizes at least t,
+and one outer bin has size of order t. Sum over the finitely many
choices of that outer endpoint and its facet. Condition on the six
-phases at $r$ (making the inner chord event measurable) and on the
-support position of the selected target facet at $s$. For any
+phases at r (making the inner chord event measurable) and on the
+support position of the selected target facet at s. For any
attained outer endpoints and supports, integrate the outer completion
-to its constant trimmed mass $F$ and bound the other band bridge
+to its constant trimmed mass F and bound the other band bridge
uniformly after dropping its extra avoidance constraints (even if it
ends on the unselected adjacent facet; Figure~fig:sewing-band).
input{figures/sewing-band}
The adjacent facet defining the selected endpoint's corner is
unconditioned: its position is independent of the selected support,
-conditional on the inner phases, and has density $O(1/g)$, since
-$V_i+A_ir/R bmod 1$ is uniform and independent of $A_i$. Each
+conditional on the inner phases, and has density O(1/g), since
+V_i+A_ir/R bmod 1 is uniform and independent of A_i. Each
attained end therefore has probability at most
-$Cmin(1,t/(1+g))$ of lying within $O(t)$ of its corner (also after
+Cmin(1,t/(1+g)) of lying within O(t) of its corner (also after
rounding). Averaging this instead of conditioning, both inner bin
-sizes are at least $t$, so the product of the two bridge bounds
+sizes are at least t, so the product of the two bridge bounds
remains uniform in the averaged parameter, giving
[
C(1+min(g,t))^{-1/2}min{1,t/(1+g)}le C(1+g)^{-1/2}.
]
-Conditional on the gap, $Xile Cl^{-1/2}$ on each window. Applying
+Conditional on the gap, Xile Cl^{-1/2} on each window. Applying
Cauchy--Schwarz to (eq:sewing-first-moment) and
(eq:sewing-second-moment) yields
-$overlineoverline EEle Cmathcal KcalK,Xi/F$ and
-$B^*ge cFoverlineoverline E/E/Xi$, proving both assertions. Averaging over
+overlineoverline EEle Cmathcal KcalK,Xi/F and
+B^*ge cFoverlineoverline E/E/Xi, proving both assertions. Averaging over
windows recombines them.
@@ -3586,65 +3568,65 @@ Corner gains and the marked-pair sum.:
The estimate above suffices for one endpoint bin. To sum over bins
with a bounded constant, retain the gain when an endpoint lies near a
-corner. Write $t=min(t_1,t_2)$, $1le tle l$. The combined per-bin
+corner. Write t=min(t_1,t_2), 1le tle l. The combined per-bin
cost before averaging over the cut gap is
-$C(1+min(g,t))^{-1/2}min{1,t/(1+g)}$. For fixed inner level $r$,
-the unnormalized gap law on the half-square $r<s$ has density at most
-$C/l$ over an interval of length $O(l)$, uniformly for the rounded
-discrete law. Splitting at $g=t$ bounds the average by
+C(1+min(g,t))^{-1/2}min{1,t/(1+g)}. For fixed inner level r,
+the unnormalized gap law on the half-square r<s has density at most
+C/l over an interval of length O(l), uniformly for the rounded
+discrete law. Splitting at g=t bounds the average by
[
frac Cl(int_0^t(1+g)^{-1/2},dg
+t(1+t)^{-1/2}int_t^{Cl}(dg)/(1+g))
le C_eta l^{-1/2}(t/l)^eta
]
-for any $0<eta<1/2$; bounded-gap atoms satisfy the same after
-increasing $C_eta$. Keeping this gain in the first sewing estimate
+for any 0<eta<1/2; bounded-gap atoms satisfy the same after
+increasing C_eta. Keeping this gain in the first sewing estimate
gives
overlineoverline EEle C_eta l^{3/4}(t/l)^etamathcal K.calK.
Indeed, the first moment on the target polygon measure is at least
-$cFoverlineoverline E$E$ and the second at most
-$CFoverlineoverline EE,l^{-1/2}(t/l)^eta$, while $Fasymp l^{-5/4}$. For
-each smaller corner bin $t$, the larger one has only
-$O(1+log(l/t))$ possible dyadic values, so their total contribution
-is at most $C_etasum_{jge0}(1+j)2^{-jeta}<∞$; bins with
-$t/lledelta$ have total coefficient tending to zero as
-$delta↓0$.
-
-Marked-pair sum and second chord moment. Let $ell_h$ denote
-the length of a chord in $D_h$, and retain chords of diameter at least
-$R/3$. Their mass is at most $CR^{3/4}$ and their first-length mass
-at least $cR^{25/12}$: every path through a sufficiently central link
+cFoverlineoverline EE and the second at most
+CFoverlineoverline EE,l^{-1/2}(t/l)^eta, while Fasymp l^{-5/4}. For
+each smaller corner bin t, the larger one has only
+O(1+log(l/t)) possible dyadic values, so their total contribution
+is at most C_etasum_{jge0}(1+j)2^{-jeta}<∞; bins with
+t/lledelta have total coefficient tending to zero as
+delta↓0.
+
+Marked-pair sum and second chord moment. Let ell_h denote
+the length of a chord in D_h, and retain chords of diameter at least
+R/3. Their mass is at most CR^{3/4} and their first-length mass
+at least cR^{25/12}: every path through a sufficiently central link
is macroscopic, and Proposition~prop:occupancy with
-Theorem~thm:nesting supplies mass at least $cR^{1/12}$ at each
-of $cR^2$ such links. The first-length mass over all chords is at
-most $CR^{25/12}$ by the same identity.
+Theorem~thm:nesting supplies mass at least cR^{1/12} at each
+of cR^2 such links. The first-length mass over all chords is at
+most CR^{25/12} by the same identity.
For the second moment, omit the initial and terminal visited centres;
-ordered pairs involving these ($O(ell_h)$ in number) are absorbed by
+ordered pairs involving these (O(ell_h) in number) are absorbed by
the first-length bound. At each remaining centre record the two
tiling-edge directions crossed by its incident links. Divide centres
into classes by triangle type, lattice translation modulo two, and
that pair of directions; the square of the total count is at most a
fixed multiple of the sum of class squares. Within a class, pairs
-with axial or zero displacement contribute at most $CRell_h$ per
-chord; their integrated contribution is at most $CR^{37/12}$.
+with axial or zero displacement contribute at most CRell_h per
+chord; their integrated contribution is at most CR^{37/12}.
For a remaining ordered pair, choose the two lattice axes spanning
the strict sector containing its displacement; at least one recorded
direction is one of those axes. Same triangle type makes the crossed
edges translates, so they supply identical axial marks used by
internal links; their displacement has even coordinates, so
-$d=P_1+P_2$ is even. Choose an integer period multiple $kP$ of length
-comparable to a large constant times $R$ so the sewn polygon projects
-injectively; then $Nasymp R$ and
+d=P_1+P_2 is even. Choose an integer period multiple kP of length
+comparable to a large constant times R so the sewn polygon projects
+injectively; then Nasymp R and
[
mathcal KcalKle CR^{-4/3}(R/d)^{15/8}.
]
-With $O(R^2)$ choices of the first centre and $O(d)$ displacements with
-$P_1+P_2=d$,
+With O(R^2) choices of the first centre and O(d) displacements with
+P_1+P_2=d,
sum_{eligible pairs}mathcal KcalK
le CR^2R^{-4/3}R^{15/8}sum_{1le dle CR}d^{1-15/8}
@@ -3653,50 +3635,50 @@ le CR^{8/3}.
Apply (eq:sewing-corner-gain) to each pair event; corner bins
sum with bounded total coefficient, and the separation factors
-$l^{3/4}$ sum geometrically up to $CR$, giving
+l^{3/4} sum geometrically up to CR, giving
overline{sum_{gamma macroscopic}ell(gamma)^2w(gamma)}
le CR^{41/12}.
-Bins with $lledelta R$ contribute at most
-$Cdelta^{3/4}R^{41/12}+CR^{37/12}$; bins with $tledelta'R$ have
-arbitrarily small relative coefficient for $delta'$ small. These
+Bins with lledelta R contribute at most
+Cdelta^{3/4}R^{41/12}+CR^{37/12}; bins with tledelta'R have
+arbitrarily small relative coefficient for delta' small. These
are estimates for endpoint restrictions, not assumed corner avoidance.
A positive fraction of long bridges.:
-Write $M_1$ for the averaged first-length mass of the macroscopic
-chords: $M_1ge c_1R^{25/12}$. By Cauchy--Schwarz, a class with
-second moment at most $ε R^{41/12}$ contributes at most
-$Csqrtε R^{25/12}$ to $M_1$. Choose $delta,delta'>0$ so
+Write M_1 for the averaged first-length mass of the macroscopic
+chords: M_1ge c_1R^{25/12}. By Cauchy--Schwarz, a class with
+second moment at most ε R^{41/12} contributes at most
+Csqrtε R^{25/12} to M_1. Choose delta,delta'>0 so
removal of close endpoints and small corner bins loses less than a
-quarter of $M_1$; chords shorter than $aR^{4/3}$ contribute at most
-$CaR^{25/12}$, less than another quarter for $a>0$ fixed small.
+quarter of M_1; chords shorter than aR^{4/3} contribute at most
+CaR^{25/12}, less than another quarter for a>0 fixed small.
The retained chords therefore have averaged first-length mass at
-least $c_2R^{25/12}$, with endpoint separation and corner distances
-all bounded below by fixed multiples of $R$, occupying a bounded
+least c_2R^{25/12}, with endpoint separation and corner distances
+all bounded below by fixed multiples of R, occupying a bounded
number of bins. In at least one bin the first-length mass has the
same lower order; Cauchy--Schwarz and (eq:chord-second-moment)
give averaged ordinary mass at least
-$cR^{50/12}/R^{41/12}=cR^{3/4}$.
+cR^{50/12}/R^{41/12}=cR^{3/4}.
The second sewing estimate extends these chords to bridges between
any prescribed pair of horizontal caps whose distances above and
-below the origin lie in $[AR,2AR]$ for fixed large $A$. With
-$lasymp R$, that estimate gives total bridge mass at least $cR^{3/4}$
-when the bottom start is free over $O(R)$ ports. Every output bridge
-contains the retained chord, hence has length at least $aR^{4/3}$.
+below the origin lie in [AR,2AR] for fixed large A. With
+lasymp R, that estimate gives total bridge mass at least cR^{3/4}
+when the bottom start is free over O(R) ports. Every output bridge
+contains the retained chord, hence has length at least aR^{4/3}.
Translation invariance of the unrestricted mass per bottom port,
together with removal of confinement and division by the number of
permitted starting ports, gives
[
B_h{ellge aR^{4/3}}ge cR^{-1/4}
]
-for every admissible cap separation $hd_0in[2AR,4AR]$. For
-arbitrary large $h$, choose $Rpropto h$ so the selected caps lie in
+for every admissible cap separation hd_0in[2AR,4AR]. For
+arbitrary large h, choose Rpropto h so the selected caps lie in
these ranges; admissible rounding changes only fixed constants.
The strip first moment and completion of the proof.:
@@ -3706,28 +3688,28 @@ It remains to prove the first-length upper bound. In the infinite
strip, summing occurrences of internal links over bridges from one
fixed bottom port equals summing the occurrence mass with free bottom
port over link representatives modulo translations along the strip;
-there are $O(h)$ such representatives. Proposition~prop:occupancy,
+there are O(h) such representatives. Proposition~prop:occupancy,
first in finite convex truncations and then by monotone exhaustion,
bounds the occurrence mass at each link by the positive
enclosing-polygon partitions at its adjacent lattice points.
-These partitions are at most $Ch^{1/12}$ even in the infinite strip:
+These partitions are at most Ch^{1/12} even in the infinite strip:
take a pure lattice period vector tilted relative to the strip's
horizontal tangent, of sufficiently large even length comparable to
-$h$, with transverse projection $>2hd_0$. The strip and its
+h, with transverse projection >2hd_0. The strip and its
translate by half this period embed disjointly in the resulting
cylinder. The disk enclosed by each polygon stays inside its convex
strip, so cannot contain the mark in the translated strip. Thus two
independent systems enclosing corresponding points inject into a
system separating the antipodal marks. The bounded-factor cylinder
-estimate of Section~sec:uniform-nesting is at most $Ch^{1/6}$,
+estimate of Section~sec:uniform-nesting is at most Ch^{1/6},
so the square of the strip partition is at most that quantity.
-Summing over the $O(h)$ link representatives,
+Summing over the O(h) link representatives,
[
sum_{gammainmathcal B_h}calB_h(ell(gamma)-1)w(gamma)le Ch^{13/12}.
]
-The one missing centre per bridge adds only $B_hle Ch^{-1/4}$,
+The one missing centre per bridge adds only B_hle Ch^{-1/4},
completing (eq:bridge-first-moment). The strip mass
(eq:bridge-mass) was proved in
Sections~c6:strip-transfer--c6:strip-asymptotics, and
@@ -3736,96 +3718,96 @@ Theorem~thm:finite-bridges is complete.
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