diff b34550f..b1f9bc4 uniform_marked_polygon_bridge_v2.txt
diff --git a/uniform_marked_polygon_bridge_v2.txt b/uniform_marked_polygon_bridge_v2.txt
index 27b9b99..6ad8d64 100644
--- a/uniform_marked_polygon_bridge_v2.txt
+++ b/uniform_marked_polygon_bridge_v2.txt
@@ -8,11 +8,11 @@
-\usepackage[microtype]
+usepackage[microtype]
-\allowdisplaybreaks[2]
+allowdisplaybreaks[2]
@@ -78,19 +78,19 @@ 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
+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
+to $v_{j+1}$. Its length is $ell(gamma)=n+1$, the
number of visited centres. Its weight is
-\[
-w(\gamma) = \kappa^{\ell(\gamma)},\qquad
-\kappa = (2\cos(\pi/8))^{-1} = 1/√(2+\sqrt 2).
-\]
-The reciprocal $\kappa^{-1}=√(2+\sqrt2)$ is the
+[
+w(gamma) = kappa^{ell(gamma)},qquad
+kappa = (2cos(pi/8))^{-1} = 1/√(2+sqrt 2).
+]
+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
+Ann. Math.~2012, proved by an exact local observable. Fix
+$kappa$. 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.
@@ -100,10 +100,10 @@ Theorem thm:finite-bridges (bridges). Take a strip
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 $\kappa^{\ell(\gamma)}$
+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
+$sum ell(gamma)kappa^{ell(gamma)}$ 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
@@ -113,7 +113,7 @@ 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
simple polygons (closed self-avoiding curves on the cylinder) that
-pass through both marked links of $\kappa^{\ell(\gamma)}$ is at
+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
@@ -123,28 +123,28 @@ direction, so it is a genuine two-point correlation bound, not a
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 $2\kappa^{\ell}$. Then the
+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 $L=\pi/8$, $b=3L$, $C=2-\sqrt2=4\cos^2 b$ are eigenvalues
+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
-$R\in d_0\mathbb Z_{>0}$ with $d_0=\sqrt3/2$, $H(y;R)$ is the
+$Rin d_0mathbb Z_bbZ_{>0}$ 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 $\mathbb H$ drawn without self-intersection
+Every closed walk of $mathbb H$bbH$ drawn without self-intersection
encloses a union of triangles. A simple polygon
-$L\subset\mathbb H$ is a simple closed curve of the honeycomb
+$Lsubsetmathbb H$bbH$ 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
@@ -153,15 +153,15 @@ unless a root is specified.
The critical weight of the self-avoiding walk is
-\kappa = (2\cos\frac\pi8)^{-1} = 1/√(2+\sqrt 2).
+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.:
@@ -170,21 +170,21 @@ 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
+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.
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 $h\ge1$, let $\mathcal B_h$ be the set of bridges of
-$\mathbb H$: paths from a fixed bottom port to any port on the
+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),
+$d_0=sqrt3/2$ is the row spacing of the triangular lattice),
staying strictly between the cuts. Put
-B_h = \sum_{\gamma\in\mathcal B_h} \kappa^{\ell(\gamma)},\qquad
-B_h(E) = \sum_{\gamma\in\mathcal B_h} \mathbf 1_E(\gamma)\kappa^{\ell(\gamma)}.
+B_h = sum_{gammainmathcal B_h}calB_h kappa^{ell(gamma)},qquad
+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
port or on the choice of pure lattice orientation. Diameters
@@ -197,14 +197,14 @@ The three main theorems.:
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_{\gamma\in\mathcal B_h} \ell(\gamma)\kappa^{\ell(\gamma)}
-&\le C h^{13/12},\\
-B_h\{\ell\ge c h^{4/3}\} &\ge c h^{-1/4}.
+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 $h\ge1$ and $x\ge0$,
+Moreover for every $hge1$ and $xge0$,
-B_h\{\diam\gamma>x\} \le C(1+x)^{-1/4}.
+B_h{diamgamma>x} le C(1+x)^{-1/4}.
@@ -216,41 +216,41 @@ 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
-$P_1,P_2\in\mathbb Z_{>0}$, $P=P_1e_1+P_2e_2$, $d=P_1+P_2$. Let
-$E\subset \mathbb H$ be a honeycomb link crossing a tiling edge
+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 $k\ge2$, quotient 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_{\gamma\ni E,\,E+P} \kappa^{\ell(\gamma)}
-\le C N^{-4/3}(N/d)^{15/8},
+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
choice of axial edge direction.
[Nesting and free boundary length]
-For a triangular lattice vertex $y$ and $R\in d_0\mathbb Z_{>0}$
-with $R\ge1$, let $H(y;R)$ be the pure-sided regular hexagon of
+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
-\[
-Z_D(y) = \sum_{\gamma_1,...,\gamma_j disjoint in D}
-2^j\,\kappa^{\sum \ell(\gamma_i)},
-\]
-the sum over systems $(\gamma_1,...,\gamma_j)$ of mutually
-disjoint simple polygons, each enclosing $y$, and $j\ge0$ (so
+[
+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 $0<a\le b<∞$ and any finite
-tiled domain $D$ with $B(y,aR)\subset D\subset B(y,bR)$,
+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
@@ -266,9 +266,9 @@ 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
and Smirnov~[cite] proved the exact
-connective constant. Beaton, Bousquet-M\'elou, de Gier,
+connective constant. Beaton, Bousquet-M'elou, de Gier,
Duminil-Copin and Guttmann~[cite] proved the
-strip-crossing mass goes to zero as $h\to∞$. 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]
@@ -280,7 +280,7 @@ 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.
-\`A~propos the analytic structure, the transfer matrix used in
+`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
@@ -300,8 +300,8 @@ 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
+chord to an exterior connector would na"ively lose a factor of
+$kappa$ 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
@@ -349,7 +349,7 @@ 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 $N\ge1$,
+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
@@ -357,140 +357,140 @@ 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 $m\to∞$ limit for its $m$-th power;
+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
-\[
-L=(\pi)/(8),\quad S=\sin(2L)=\frac 1{\sqrt2},\quad b=3L,\quad
-\gamma=\frac b2,\quad \kappa=(2\cos L)^{-1}=1/√(2+\sqrt2).
-\]
-We write $t_j=\sin(u+jL)$ when the argument $u$ is clear from context.
+[
+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).
+]
+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} = 2\cos L\, t_j = t_j/\kappa,\qquad
+t_{j-1}+t_{j+1} = 2cos L, t_j = t_j/kappa,qquad
t_{j-2}t_{j+2} = t_j^2 - S^2,
-both direct consequences of $\sin(a+b)+\sin(a-b)=2\sin a\cos b$ 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.:
-Let $V=\mathbb C^3$ have basis $\{|+>,|0>,|->\}$,
-interpreted as spins $+1,0,-1$. Set $s = e^{2iL} = e^{i\pi/4}$ so
-$s^4=-1$ and $s^2+s^{-2}=0$. Put $Q(u)=\sin(2L+u)\sin(b+u)$ and
+Let $V=mathbb C^3$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
define
-a(u)&=(S\sin(b-u))/(Q(u)),&
-h(u)&=(S\sin u)/(Q(u)),&
-v(u)&=(\sin(b-u)\sin u)/(Q(u)),\\
-c(u)&=(\sin(b-u)\sin(2L-u))/(Q(u)),&
-d(u)&=(\sin u \sin(u-L))/(Q(u)).
+a(u)&=(Ssin(b-u))/(Q(u)),&
+h(u)&=(Ssin u)/(Q(u)),&
+v(u)&=(sin(b-u)sin u)/(Q(u)),
+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),\\
-&R(+0,+0)=R(0+,0+)=R(-0,-0)=R(0-,0-)=v(u),\\
-&R(+0,0+)=R(0+,+0)=R(-0,0-)=R(0-,-0)=a(u),\\
+&R(++,++)=R(--,--)=c(u),
+&R(+0,+0)=R(0+,0+)=R(-0,-0)=R(0-,0-)=v(u),
+&R(+0,0+)=R(0+,+0)=R(-0,0-)=R(0-,-0)=a(u),
&
-R(+-,+-)&R(+-,00)&R(+-, -+)\\
-R(00,+-)&R(00,00)&R(00,-+)\\
+R(+-,+-)&R(+-,00)&R(+-, -+)
+R(00,+-)&R(00,00)&R(00,-+)
R(-+,+-)&R(-+,00)&R(-+,-+)
=
-d(u)&h(u)/s&c(u)+d(u)/s^2\\
-s\,h(u)&1&h(u)/s\\
-c(u)+d(u)s^2&s\,h(u)&d(u)
+d(u)&h(u)/s&c(u)+d(u)/s^2
+s,h(u)&1&h(u)/s
+c(u)+d(u)s^2&s,h(u)&d(u)
.
-All other entries vanish: $R(\pm\pm,\mp\mp)$, $R(0\pm,\pm-)$, etc.\ are
+All other entries vanish: $R(pmpm,mpmp)$, $R(0pm,pm-)$, etc. are
zero. Summation over intermediate spins is implied.
Structural identities of $R$.:
Let $P$ denote the swap $P(a ⊗ b)=b ⊗ a$,
-$J=diag(-1,1,-1)$, $F=(0&0&1\\0&1&0\\1&0&0)$
+$J=diag(-1,1,-1)$, $F=(0&0&10&1&01&0&0)$
(spin flip), and
-Ω=\sum_{p\in\{-,0,+\}} s^{-p}\,|p,-p>\in V ⊗ V.
+Ω=sum_{pin{-,0,+}} s^{-p},|p,-p>in V ⊗ V.
-Define moreover the rank-two operator $W\inHom(V,V ⊗ V)$ by
+Define moreover the rank-two operator $WinHom(V,V ⊗ V)$ by
-W(|+>)=\kappa(|+,0>+|0,+>),\quad
-W(|->)=\kappa(|-,0>+|0,->),\quad
-W(|0>)=|00>+\kappa\sum_{p\in\{-,+\}} s^{-p}|p,-p>,
+W(|+>)=kappa(|+,0>+|0,+>),quad
+W(|->)=kappa(|-,0>+|0,->),quad
+W(|0>)=|00>+kappasum_{pin{-,+}} s^{-p}|p,-p>,
-and $Z=W^*=\bar W^T\inHom(V ⊗ V,V)$.
+and $Z=W^*=barbar W^TW^TinHom(V ⊗ V,V)$.
-\paragraph*{The identities.} We use the following, verified by direct
+paragraph*{The identities.} We use the following, verified by direct
substitution in (eq:R-coeffs)--(eq:R-matrix).
[$R$-matrix identities]
For all complex $u$ away from poles,
-R_{21}(u) &= R_{12}^T(u),&(transpose)\\
-R_{12}(u) R_{12}^T(-u) &= I,&(unitarity)\\
-R_{12}(u+\pi) &= J_1 R_{12}(u) J_1,&(pi-periodicity)\\
-R_{12}(b-u)_{ij,kl} &= s^{k-i} R_{12}(u)_{-i,l;-k,j},&(crossing)\\
-R_{12}(0) &= P_{12},&(permutation at zero)\\
-R_{12}(b) &= Ω\,\barΩ^T,&(rank-one at b)\\
+R_{21}(u) &= R_{12}^T(u),&(transpose)
+R_{12}(u) R_{12}^T(-u) &= I,&(unitarity)
+R_{12}(u+pi) &= J_1 R_{12}(u) J_1,&(pi-periodicity)
+R_{12}(b-u)_{ij,kl} &= s^{k-i} R_{12}(u)_{-i,l;-k,j},&(crossing)
+R_{12}(0) &= P_{12},&(permutation at zero)
+R_{12}(b) &= Ω,barbarΩ^T,&(rank-oneΩ^T,&(rank-one at b)
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,
+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
+$sin$-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.
+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 $u\mapsto
-u+\pi$, matching the $J$ insertion on either side; on $(+-,00,-+)$,
+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
-a(b-u)&=S\sin(u)\big/Q(u)=a(u),\quad
-h(b-u)=S\sin(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...
+a(b-u)&=Ssin(u)big/Q(u)=a(u),quad
+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
+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 $k-i$.
+$s$, $s^{-1}$, $s^{pm2}$ 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 $u=b$: $Q(b)=\sin(4L)\sin(2b)
-= 1 \cdot \sin(6L) = \sin(\pi/4+\pi/2) = \sin(3\pi/4) = S$;
-direct check gives $R(b)=Ω\barΩ^T$. At $u=2L$:
-$Q(2L)=\sin(4L)\sin(2L+3L)=\sin(4L)\sin(5L)=1\cdot S$; direct check
+$v=c=1$; $R$ 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 $R(2L)=WZ$.
@@ -502,7 +502,7 @@ The key insertion identity:
[Basic fusion]
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).
+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 $f ⊗ V_3$.
@@ -511,47 +511,47 @@ acts on $f ⊗ V_3$.
Proof.
Transposing gives the adjoint form
-Z_{12}\,R_{13}(u+L)\,R_{23}(u-L) = R_{f3}(u)\,Z_{12}.
+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
(eq:R-unitarity)). The required identity becomes
-\[
-R_{23}(u-L)\,R_{13}(u+L)\,W_{12}\,R_{32}(L-u) = W_{12}.
-\]
+[
+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
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.
The pairs of spins we need to consider are (up to spin flip): $(+,+)$
-and $(+,-)$ and $(+,0)$ and $(0,0)$ and $(+,f)$-type with $f\in\{+,0,-\}$.
-Each case reduces to sine-product identities in $t_j=\sin(u+jL)$. The
+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:
+$sin(cdot)$ products), is:
On $(+,+)$-input:
-cv' = c_+,\quad cc'=v_+,\quad ca'=a_+.
+cv' = c_+,quad cc'=v_+,quad ca'=a_+.
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_+,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_+.
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_+,quad v'+a'h=v_+,quad h'+kappa h c'=kappa a_+.
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_+,quad hh'+kappa(cd'+c'd)=0,quad
+dv'+ha'=d_+,quad da'+hv'=0,
together with the $(+,+)$-input relations applied to the $(-,-)$ pair.
@@ -559,16 +559,16 @@ 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_{-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_2t_1 - t_{-2}t_{-1}=t_0t_4/\kappa.
+t_{-2}t_2 = t_0^2 - S^2,quad
+t_{j-1}+t_{j+1}=t_j/kappa,quad
+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_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
+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.
@@ -579,7 +579,7 @@ The Yang--Baxter equation.:
[Yang--Baxter]
For all $u,v$,
-R_{12}(u-v)\,R_{13}(u)\,R_{23}(v) = R_{23}(v)\,R_{13}(u)\,R_{12}(u-v).
+R_{12}(u-v),R_{13}(u),R_{23}(v) = R_{23}(v),R_{13}(u),R_{12}(u-v).
@@ -591,14 +591,14 @@ 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 $R(b)=Ω\barΩ^T$: $W$ now inserts $Ω$, and
+$u-v=b$, 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
-$u\in\{v,v+2L,v+b\}$ therefore determine the entry. This proves
+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.
@@ -610,13 +610,13 @@ 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
-k(t)=(\sin(\gamma-t))/(\sin(\gamma+t)),\qquad
-l(t)=(\sin(2t))/(\sin(\gamma+t)),
+k(t)=(sin(gamma-t))/(sin(gamma+t)),qquad
+l(t)=(sin(2t))/(sin(gamma+t)),
-$D_e(t)=diag(k(t),1,k(t))+ e\,l(t)(E_{+,0}+s E_{0,-})\in
+$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),qquad K_e(t)_{,p,-m}
=s^m(D_e(t))_{pm}.
That is, $K_e(t)$ makes the $+/-$ boundary identifications via the $s$
@@ -626,92 +626,92 @@ weight while imposing $D_e$ on the surviving pair.
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),\\
-D_{e,2}(t-L)\,R(2t)\,D_{e,1}(t+L)\,\bar W &= W\,D_e(t),\\
-W^T D_{e,2}(t+L)\,R(2t)\,D_{e,1}(t-L) &= D_e(t)\,\bar W^T.
+D_e(t)D_e(-t) &= I,
+R(t-r) K_e(t) &= P K_e(r),
+D_{e,2}(t-L),R(2t),D_{e,1}(t+L),barbar WW &= W,D_e(t),
+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
-$l(-t)=\sin(-2t)/\sin(\gamma-t)=-\sin(2t)/\sin(\gamma-t)
-=-l(t)\sin(\gamma+t)/\sin(\gamma-t)=-l(t)/k(-t)$;
+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
+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 $\barΩ$), the
+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
+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)=S\cos y,\qquad
-\cos(L\pm y)=\sin(b\mp y).
+sin(L-y)sin(b-2y)+sin(L+y)sin(b+2y)=Scos y,qquad
+cos(Lpm y)=sin(bmp y).
Fusion below. We sketch. Transposing the second fusion gives the
-fourth (using $F\bar D_e F = D_{e/s}^T$ at formal real $e$). For 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
-$D_{e,2}R(2t)D_{e,1}\bar W$ the constant term in $e$ reduces to
+$Q(2t)=sin(4L)sin(2t+3L)cdot...$; 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
-$H_\pm=\sin(\gamma\pm t)$, $G_\pm=\cos(\gamma\pm t)$,
-$F_\pm=\sin(L/2\pm t)$, $P_\pm=\sin(2L\pm 2t)$, $T_0=\sin(2t)$,
+$k_-(h+kappa c k_+)=kappa$. 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
-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
-H_-=\kappa(F_-+G_+),\\
-S+\sin(2t-L)=2F_+G_-,\quad
-P_+ H_+ G_+-P_- H_- G_-=\cos L\,T_0\cos(2t).
+P_+ G_+-S G_-=T_0 F_-,quad
+P_+F_+-S F_-=T_0 G_-,quad
+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
+H_-=kappa(F_-+G_+),
+S+sin(2t-L)=2F_+G_-,quad
+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 $T(t)$.:
-For an integer $N\ge1$, 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}
+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 $\alpha\in\{1,2\}$ labels the two rows and $\zeta_j$ are the
+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
-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},qquad r=b-t,
-where $e_\ell$, $e_r$ are formal nilpotent parameters ($e_\ell^2=e_r^2=0$).
+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.
[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.
+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.
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
@@ -720,11 +720,11 @@ $T$ 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
-$K_e(0)=\barΩ$. Both local $R$'s, $R(0)=P$ and $R(b)=Ω\barΩ^T$,
-act as rank-one contractions on the input $\barΩ$, giving
+$K_e(0)=barbarΩ$.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
+$T(4L)$: 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.
@@ -732,7 +732,7 @@ closes the contraction, giving identity.
Commutation of $T(t)$ and $T(y)$.:
[Commuting double rows]
-For all $t,y$ in $\mathbb C$ away from poles,
+For all $t,y$ in $mathbb C$bbC$ away from poles,
T(t)T(y) = T(y)T(t).
@@ -740,7 +740,7 @@ 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
+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
@@ -756,14 +756,14 @@ 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), $R(0)=P$,
-$R(b)=Ω\barΩ^T$ or $R(2L)=WZ$. For instance at $t=y$ both sides
+$R(b)=ΩbarbarΩ^T$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 $D_{e,2}(y)R(t+y)K_e(t) = PD_{e,1}(y)K_e(-y)=Ω$.
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.:
@@ -773,11 +773,11 @@ The key algebraic step in extracting the physical eigenvalue is the
following.
[Row fusion]
-For all $t$ and all $i\in\{1,...,N\}$,
+For all $t$ and all $iin{1,...,N}$,
-T(t-L) T(t+L) = T(t)\quadon the physical sector.
+T(t-L) T(t+L) = T(t)quadon the physical sector.
-More precisely, $T(t)^m_{∅,∅}=1$ for all $m\ge0$
+More precisely, $T(t)^m_{∅,∅}=1$ for all $mge0$
and all $t$, and the above identity holds on the total-spin-zero
eigenline of $T(t)$.
@@ -785,8 +785,8 @@ 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
+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 $R_{f3}$
@@ -802,13 +802,13 @@ The physical eigenvalue at homogeneous rapidities.:
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
+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
@@ -816,14 +816,14 @@ 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) & $d(u)$ \\
+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
@@ -834,148 +834,148 @@ 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
-$\pm 2\pi$ (contractible) or $0$ (essential, wrapping around the
+$pm 2pi$ (contractible) or $0$ (essential, wrapping around the
period). Summing the two orientations:
-\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.
+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
contribution).
Physical tile weights.:
-At $t=\gamma$ we have $k(\gamma)=\sin(\gamma-\gamma)/\sin(2\gamma)=0$:
-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)&=(S\sin(2L))/(\sin(3L))=\frac{1}{√(2+\sqrt2)}=\kappa,\\
-h(L)&=v(L)=c(L)=(S\sin(L))/(\sin(3L))=(1)/(2+\sqrt2)=\kappa^2,\\
-d(L)&=(\sin(L)\sin(0))/(\sin(3L))=0.
+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 $\kappa^2=\kappa\cdot\kappa$.
+triangle centre costs $kappa$, and pairing two arcs at a shared centre
+costs $kappa^2=kappacdotkappa$.
-The $m\to∞$ limit.:
-Entries of $T(\gamma)^m$ are sums over configurations in an $m$-long,
+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 $C<1$
+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 $m\to∞$. Surviving configurations are unions of
+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
+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).
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 $m\to∞$ limiting
+identity in Section~sec:sewing). So the $mto∞$ limiting
empty-to-empty entry of the normalized projection
-$P_0 := \lim_{m\to∞}T(\gamma)^m$ 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
+$T(gamma)$ 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 $1+\xi s^{-6p}+\xi^{-1}s^{6p}\neq 1$),
+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
+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
+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
+in $e_ell e_r$, is
-T(t) on the eigenvalue-$1$ line\quad↝\quad
-1+e_\ell\,e_r\,q(t;\xi)+O((e_\ell e_r)^2)
+T(t) on the eigenvalue-$1$ linequad↝quad
+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
+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).
+[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 coefficient $q$ is the bridge mass.:
-At $t=\gamma$ and $\xi=-L/2$ (homogeneous), the coefficient $q$
+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
-B_N = (1)/(2)q(\gamma;\xi)
+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_{m\to∞}(1)/(2m)
-[e_\ell e_r](T(\gamma)^m)_{∅,∅}
-=(1)/(2)q(\gamma;\xi).
+B_N = lim_{mto∞}(1)/(2m)
+[e_ell e_r](T(gamma)^m)_{∅,∅}
+=(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 $\mathbf X(t)=\cos(2t)$.
+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 $\mathbf X(t)$ of degree at most $N+1$
+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)^{N}(\mathbf X+\cos 2L)^{N+1}},
+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 $\mathbf X\to∞$ is $4\mathbf X+\rho$.
+and the polynomial part of $q$ as $mathbf XbfXto∞$ is $4mathbf X+bfX+rho$.
Proof.
-The $m\to∞$ limit and (1) column-by-column: the only pole sources
+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 $\mathbf X=\cos(2t)$ converts all $t$-dependences to
-rational in $\mathbf X$ since $\sin(2t)$ etc.\ are polynomial in
-$\cos(2t)$. The two boundary rows give poles at $\mathbf X=-1$ and
-$\mathbf X=-\cos(2L)$; the $N$ local poles pile at
-$(\mathbf X+1)^N(\mathbf X+\cos 2L)^{N+1}$. (The extra $\cos 2L$
+$K$'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 $\mathbf X\to∞$, $T(t)$ tends to
-$I+$ single-$t$-matrix terms by unitarity, contributing $4\mathbf X$
-from the leading coefficient; a fixed scalar $\rho$ traces the linear
+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
correction.
[Zeros of $q$]
The following hold:
-q(3L/2;\xi)=q(5L/2;\xi)=0.
+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)
+q(gamma+u-L;xi)+q(gamma+u+L;xi)-q(gamma+u;xi)
has a zero of multiplicity
-\ge 2N & at u=0,\\
-\ge N & at u=L.
+ge 2N & at u=0,
+ge N & at u=L.
@@ -983,19 +983,19 @@ q(\gamma+u-L;\xi)+q(\gamma+u+L;\xi)-q(\gamma+u;\xi)
Proof.
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.
+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 $b=3L$, $\gamma=3L/2$:
-\[
-p(3L/2)=q(3L/2+\gamma)=q(3L)=q(b)=0,\qquad
-p(5L/2)=q(5L/2+\gamma)=q(5L/2+3L/2)=q(4L)=0.
-\]
+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(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 $t=\gamma$
+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).
@@ -1004,85 +1004,85 @@ matrix identities, the difference vanishes to order $2N$ at $t=\gamma$
The scalar crossing function.:
-At homogeneous $\xi_j$ (i.e., at the physical rapidities with correction
+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)=\cos(2(\gamma+u)).
+p(u)=q(gamma+u;xi),qquad 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 $\mathbf X$, and
+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))}{(\mathbf X+1)^N(\mathbf X+\cos 2L)^{N+1}}.
+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 $P(\mathbf X(3L/2))=P(\mathbf X(5L/2))=0$, from
+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
+item 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 L\!((C_u)/(D)),
+p(u)=mathcal LcalL!((C_u)/(D)),
where
-\[
-C_u(x)=cosh(u\sqrt x),\quad D(x)=2C_L(x)-1,
-\]
-$\mathcal L$ is a fixed linear functional, and $x=z^2$.
+[
+C_u(x)=cosh(usqrt x),quad D(x)=2C_L(x)-1,
+]
+$mathcal L$calL$ is a fixed linear functional, and $x=z^2$.
-The value $2B_N=2\cdot(1)/(2)q(\gamma)=q(\gamma)=p(0)$ becomes
+The value $2B_N=2cdot(1)/(2)q(gamma)=q(gamma)=p(0)$ becomes
-2B_N=p(0)=\mathcal L(1/D).
+2B_N=p(0)=mathcal L(1/D).calL(1/D).
The finite functional.:
-We now pin down $\mathcal L$. 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 $\mathcal L$ on the span of
+There is a unique linear functional $mathcal L$calL$ on the span of
$M$, $C_{L/2}$, $C_{3L/2}/D$ such that:
-\mathcal L(f) &= 0\ for every f\in M,\\
-\mathcal L(C_{L/2}) &= 0,\\
-\mathcal L(C_{3L/2}-C_{L/2}/D) &= 0.
+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, $2B_N=\mathcal L(1/D)$ 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 $\mathbf X$. 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)]=(S\sin(2m))/(\mathbf X+\cos 2m)
+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 $\mathcal L(1/D)=p(0)=2B_N$.
+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
-$(\mathbf X+1)^{-N}(\mathbf X+\cos 2L)^{-(N+1)}$ 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 $z=2ik$, $k\ge1$, 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.
@@ -1090,74 +1090,74 @@ sums to the displayed $\tan$-combination.
The positive interpolation estimate.:
Section~c6:strip-transfer reduced the strip bridge mass to a
-single linear functional $\mathcal L$ on a $(2N+2)$-dimensional space.
-Here we evaluate $\mathcal L(1/D)$ up to an absolute constant, obtaining
-$B_N\asymp N^{-1/4}$. The engine is a determinant-density representation
+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
and an asymptotic estimate of characteristic products.
Spaces, measure, determinant density.:
Retain from Proposition~prop:finite-functional:
-\item measure $d\mu(x)=zD(x)\,dz/sinh(4Lz)$, $x=z^2$;
-\item $M_N=span\{1,x,...,x^{N-1};∂_m^jC_m(x)|_{m=L}\}_{j=0}^{N-1}$;
-\item $\mathcal L(M_N)=\mathcal L(C_{L/2})=
-\mathcal L(C_{3L/2}-C_{L/2}/D)=0$;
-\item $2B_N=\mathcal L(1/D)$.
+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)$.
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(L\sqrt x)/\sqrt x,\\
-F_0&=M_N+\mathbb Cc_1,\quad F=F_0+\mathbb Cc_3,\quad G=M_N+\mathbb Cg.
+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.
-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
-\nu = \frac{\det F_0(X)\det G(X)\prod_{i=1}^d d\mu(x_i)}
-{\int\det F_0\det G\prod d\mu}.
+nu = frac{det F_0(X)det G(X)prod_{i=1}^d dmu(x_i)}
+{intdet F_0det Gprod dmu}.
By Lemma~lem:jet-positivity below this is a probability density.
[Jet positivity]
-Enumerate "labels" $\beta\in\{0,L^2\}$ each with multiplicity $N$, plus
+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 $C_{\sqrt\beta}(x)$ at the labels is positive; at
+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.
-$C_{\sqrt\beta}(x)=\sum_{j\ge0}\beta^jx^j/(2j)!$ 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
-$\beta\ge0$) gives the ordered jet determinants; the $g$ row is a
+$betage0$) gives the ordered jet determinants; the $g$ row is a
positive integral of $C_t$ rows, $0<t<L$.
[Determinant density]
-For $H$ with $\int|Hf|d\mu<∞$ ($f\in M_N+\mathbb Cg$):
+For $H$ with $int|Hf|dmu<∞$ ($fin M_N+mathbb Cg$):bbCg$):
-(\mathcal L(H))/(4a_*)=\nu[e_{-4}H]+s_*\,\nu[e_0H],
+(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 $\mathcal L$.
+at nodes $X$, and $s_*=rho/(4a_*)$ the scalar in $mathcal L$.calL$.
Proof.
-Apply Andr\'eief's identity
-$\int\det F_0\det G\prod d\mu=\det(\int F_0G^Td\mu)$ row by row: the
-integral $\int Hf\,d\mu$ 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 $\mathcal L$
+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$
component. See Tracy--Widom [cite] for the identity
itself.
@@ -1165,88 +1165,88 @@ itself.
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],
+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.
Also define the tilted measure
-d\nu_3(X)\propto -e_0c_3(X)\,d\nu(X).
+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}.
+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 $1\le r\le2$)
-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
-$\det\int V B^T(x-\lambda)d\mu'$ for either $(F_0,G_t)$ vs $(M,M)$,
-or $(A,G_t)$ vs $(U,U)$ (here $A=\{f\in F:f(0)=0\}$, $U=\{f\in M:f(0)=0\}$).
+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 $m$
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 $\mathcal P_{<N}$ and $\mathcal P_{<CN}$ for
+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\,d\mu_c=(1)/(1-w_mw_n),\quad w_m=\tan(\alpha m).
+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).
(The identity follows from
-$\int C_mC_nd\mu_c=\frac12(\sec\alpha(m+n)+\sec\alpha(m-n))$, 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
-$K_{n,W'}(v)=\sup_{\deg h\le n}|h(v)|^2/\int|h|^2dW'$ is the
+$K_{n,mu_c}(-v^2)asymp n^{2cv/pi}$ 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}.
+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 $(x+z^2)x^j$, $0\le j<n$.
+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 $r'=r\mu/\mu$, the square-root asymptotic
+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.
Interpolation residuals.:
[Residual bound]
-Let $\lambda_*=\pi/(3L)=8/3$. For $A\in\{e,E\}$ and $a\in\{0,4\}$ and
-$j\in\{0,1\}$, 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:
-0\le A_{-a}h_j\le C\prod_{i=1}^N(v_i+a)/(v_i+\lambda_*^2)
+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.
+$prod_{i=1}^d$ also holds.
Proof. [Sketch]
@@ -1258,16 +1258,16 @@ space is a convex combination of monomial residuals with powers
$0,...,N-1$ 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
-$\theta_N=(\sum\beta_i)/(2(2N+1))$; for large $N$ the minors remain
+$theta_N=(sumbeta_i)/(2(2N+1))$; for large $N$ the minors remain
positive after translating by $a$.
-Laplace structure. Write $h_j(w-a)=h_j(-a)\mathbb E e^{-wZ_j}$,
-where $Z_j\ge0$ is a random variable. Explicitly, the product formula
+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
-$D(x)\propto(x+\lambda_*^2)\prod_{k\ge2}(x+s_k)$ with
-$s_k=((2\pi k+\pi/3)/L)^2$; similarly $c_1$ and $(c_3-c_1)$ have
+$D(x)propto(x+lambda_*^2)prod_{kge2}(x+s_k)$ with
+$s_k=((2pi k+pi/3)/L)^2$; similarly $c_1$ and $(c_3-c_1)$ have
product form. Hence
-$\mathbb P(Z_j\ge t)\asymp e^{-\lambda_*^2t}$ (for $h_1$, one pairs
+$mathbb P(Z_jbbP(Z_jge t)asymp e^{-lambda_*^2t}$ (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).
@@ -1277,9 +1277,9 @@ 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:
-\[
-\mathbb E[e^{-(\lambda_*^2-a)Y}]=\prod_{i}(v_i+a)/(v_i+\lambda_*^2)
-\]
+[
+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 $v_i+a$.
@@ -1287,103 +1287,103 @@ 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_{-4}h_i]|=O(1),
+nu[-e_0c_3]gtrsim N^{-1/2},quad
+|nu[e_0h_i]|le N^{-1+o(1)},quad
+|nu[e_{-4}h_i]|=O(1),
and the tilted target
-\nu_3[E_{-4}h_0]\asymp N^{-1/4}.
+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 $\mu_b\propto dx/cosh(b\sqrt x)$ 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=L\sqrt x,
+phi(x)=(z(cosh(4z)+cosh(2z)-cosh(3z)))/(sinh(4z)),quad 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 $(0,∞)^d$:
-$\nu_b[f]\le C\nu[f]$.
+$nu_b[f]le Cnu[f]$.
Proof.
-A probability measure $\eta$ on $(0,∞)^d$ is $MTP_2$ if its
-density $w$ satisfies $w(x\wedge y)w(x\vee y)\ge w(x)w(y)$ for all
-$x,y$, with $x\wedge y$ and $x\vee y$ 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 $w=\det(F,G)\prod_i d\mu(x_i)$,
-$MTP_2$ 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 $2\times 2$ 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 $d\mu$ and $d\mu_b$, are therefore $MTP_2$.
-
-Step 2: FKG. For an $MTP_2$ measure $\eta$ and any
+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 $f_1,...,f_m$,
-$\eta[\prod_jf_j]\ge\prod_j\eta[f_j]$ (Harris). Indeed one-coordinate
+$eta[prod_jf_j]geprod_jeta[f_j]$ (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
+ratio in the other coordinates by Pl"ucker's quadratic relation, so
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,
-$d\nu_b/d\nu=\prod_i\phi(x_i)$ with $\phi$ the increasing
+$dnu_b/dnu=prod_iphi(x_i)$ with $phi$ the increasing
one-variable ratio displayed above, whose Taylor coefficients of
-$\phi\log z$ (recall $z=L\sqrt x$) 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
-$\nu_b[f]\le C\nu[f]$ for bounded coordinatewise increasing $f$ and
+$philog z$ (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.
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
+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'),
+(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
-$A_y=\nu[e_yc_3]$, $H_{j,y}=\nu[e_yh_j]$
-($y\in\{0,-4\}$). The constraint $\mathcal L(c_3-h_1)=0$ 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 $mathcal L(c_3-h_1)=0$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
+$|H_{1,0}|le N^{-1+o(1)}$.
-The exact form of $\mathcal L(h_0)/(4a_*)$ is then
+The exact form of $mathcal L(h_0)/(4a_*)$calL(h_0)/(4a_*)$ is then
-(\mathcal L(h_0))/(4a_*)
-=\frac{A_0\,\nu_3[E_{-4}h_0]
+(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_0\nu_3[E_{-4}h_0]\asymp N^{-1/4}\cdot
+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
-$\mathcal L(h_0)/(4a_*)\asymp N^{-1/4}$, hence
-$B_N\asymp N^{-1/4}$.
+$mathcal L(h_0)/(4a_*)calL(h_0)/(4a_*)asymp N^{-1/4}$, 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
@@ -1405,51 +1405,51 @@ 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 $\mathcal Z_N(y)$ sum systems of
+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
+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
-$C=2-\sqrt2=4\cos^2 b$ and
+$C=2-sqrt2=4cos^2 b$ and
-R_s(d)=(-1)/(\sin(d+L)\sin(d-L)),\quad R_c(d)=\cos^2 d,\quad
-D(d)=\sin(d+2L)\sin(d-2L)\sin(d+b)\sin(d-b).
+R_s(d)=(-1)/(sin(d+L)sin(d-L)),quad R_c(d)=cos^2 d,quad
+D(d)=sin(d+2L)sin(d-2L)sin(d+b)sin(d-b).
-Sum over site labels $\alpha_j\in\{+1,0,-1\}$, with total sum in the
-two groups $\{1,...,m\}$ and $\{m+1,...,N\}$ equal to a common $h$.
-Weight every configuration by $y^h\prod_jc_{\alpha_j}$ 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_{\alpha\alpha'}(v_j-v_i)=
-1,&(\alpha,\alpha')\in\{(+,+),(-,-)\},\\
-H(v_j-v_i),&(\alpha,\alpha')=(0,0),\\
-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')=(+,-),
+P_{alphaalpha'}(v_j-v_i)=
+1,&(alpha,alpha')in{(+,+),(-,-)},
+H(v_j-v_i),&(alpha,alpha')=(0,0),
+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 $P_{\alpha\alpha'}(d)=P_{\alpha'\alpha}(-d)$, and with $H=R_s$
+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}^m\alpha_j=\sum_{m+1}^N\alpha_j}
-y^{\sum\alpha_j}\prod_jc_{\alpha_j}\prod_{i<j}P_{\alpha_i\alpha_j}(v_j-v_i).
+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).
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),qquad h(u)=(cos u)/(B(u)).
-Let $\mathsf h_N$ be the skew matrix with upper entries
-$(\mathsf h_N)_{jk}=h(v_k-v_j)$, $j<k$; for odd $N$ append a last
+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
+Set $a_{2l}=2^l/sqrt 2$, $a_{2l+1}=(-2)^l$, and
-f_N(v)=a_N\prod_{j<k}B(v_k-v_j)\,Pf\,\mathsf h_N.
+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'}.
+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.
@@ -1457,7 +1457,7 @@ The apparent singularities in~(eq:f_N) are removable
[Homogeneous physical pairing]
For $y>0$,
-\mathcal Z_N(y)=\left.(G_N(v))/(2f_N(v)^2)|_{v_1=...=v_N=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 $y$.
@@ -1469,44 +1469,44 @@ family of fusion conditions.
Periodic eigenlines.:
-Let $K(\xi)_{pp}=\xi^p$ denote the spin twist ($p\in\{+1,0,-1\}$)
-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),
-\qquad\xi=i\ or -i,
+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
action is from the right.
[Polynomial vacuum]
-For $N\ge1$ and generic $(v_j)$, $T_i(t)$ has a simple common eigenline
+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 $v_j$,
+$w^i_∅=1$, 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 $v_j\mapsto v_j+\pi$.
-\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_{z\in U}\cos(z-u-L)\cdot W_{12}\Psi_{N-1}(u+L,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)=2\cos b\prod_{z\in U}\cos(z-u-L)\cos(z-u-2L)\cdotΩ_{12}\Psi_{N-2}(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 $|v_N|\to i∞$,
-$\Psi_N(v)/\prod_{j<N}\cos(v_N-v_j)\to\sqrt2\Psi_{N-1}(v_1,...,v_{N-1}) ⊗ |0>_N$.
-\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,quad T_0T_3=I,quad 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
@@ -1514,111 +1514,111 @@ $1$ (loop cancellation, as in Section~sec:strip-physical).
Judicious imaginary-limit sampling reduces the simplicity on total spin
zero to a base case $N=1$.
-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
+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
-$v_q-v_p\in\{0,2L,b\}$, the local $R_{pq}$ and its fusions act as stated.
+$v_q-v_pin{0,2L,b}$, 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
-\[
-T_\xi(t)=I+\sum_k[A_k\cot(t+v_k+2L)+B'_k\cot(t+v_k+3L)],
-\qquad\sum_k(A_k+B'_k)=0.
-\]
-Applying the fusion equations to $T_j=T_\xi(-v_p+jL)$ gives commuting-operator
-annihilator rows $\mathcal A_{p,0}=c_0-c_1+c_2+X_0c_2$ and
-$\mathcal A_{p,1}=c_1-c_2+c_3+X_1c_3$. Collecting into operator
-matrix $\mathcal A$, one has
-$\det\mathcal A=(\det H_{\mathcal A})P^\xi$, where $H_{\mathcal A}$
-is the cotangent kernel $r(u)=\cot(u+3L)-\cot(u+4L)+\cot(u+5L)$.
-The scalar Pfaffian $J_N=Pf(H_{\mathcal A})\prod_{j<k}p(v_k-v_j)$,
-
-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.
+[
+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 $J_N=Pf(H_{mathcal A}calA)prod_{j<k}p(v_k-v_j)$,
+
+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.
Next, the two-site computation
-J_2=\cos u\,[S t_*-(t_*^2/2)\cos 2u],\qquad t_*=-(1)/(1+S),
+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 $u\equiv\pi/2$,
-and two complex from $\cos 2u=-1-\sqrt2$. This gives generic simplicity
-at $v_j-v_k\in\{L,2L\}$. Combining with
-fusion via $W$ and $Ω$, the total pole degree of $P^\xi$ is
-bounded by $3(N-1)$, hence $d_N^i+d_N^{-i}\le 3(N-1)$.
+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 $(v_1,v_2,v_3)=(u,u+2L,u+5L)$
+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 $z=0$, $\Psi\inim(W_{12})$; on $z'=z$ and $z'=0$ we get
+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
+$K_3 J^{rm spin}_1 Ω_{31}$ (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 $\ge2$.
+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
+(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 $f_N=\Psi_N^∅$.
+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 $v_2-v_1\in\{b\}$ in the double fusion~(eq:psi-fusion-b):
-the wheel factor there forces $d_N^i\ge N-1$ and $d_N^{-i}\ge 2(N-1)$.
+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_1\prod_{z\in U}\cos(z-u-L)\cdot W_{12}\Psi_{N-1}(u+L,U),\\
-\Psi_N(u,u+b,U) &= c_2\prod_{z\in U}\cos(z-u-L)\cos(z-u-2L)\cdotΩ_{12}\Psi_{N-2}(U),
+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 $c_2=2\cos b$
+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
-$Z(v)=\ell^{-i}(v)(\prod_{j\le m}K_j(\delta))w^i(v)$,
-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:
-\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 $U$).
+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
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:
+Evaluating at $v_1=...=v_N=0$ with the sandwiched $K_j(delta)$'s:
-\item $\mathcal Z_N$ at homogeneous $v$ has a natural expansion as an
+item $mathcal Z_N$calZ_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 $(-1)^{1_{A\in D'}}$ plus
-$\delta^{1_{A\in D'}-1_{B\in D'}}$ from the sandwich. Reversing
+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
+item The tile weights in $G_N$ without phases are honeycomb path
weights.
-\item The empty system gives $1$, matching $(∅,∅)$
+item The empty system gives $1$, matching $(∅,∅)$
entry.
-For positive $y$, exhaustion proves $\mathcal Z_N=G_N/(2f_N^2)|_{v=0}$,
+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.
@@ -1630,24 +1630,24 @@ 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.:
Put
-P_m(e^r)=\frac{G(e^r)}{C^mD(0)^{m(m-1)}}.
+P_m(e^r)=frac{G(e^r)}{C^mD(0)^{m(m-1)}}.
-With $L=\pi/8$, $b=3L$, $C=2-\sqrt2$, $\widehat\rho(p)=1/(2cosh Lp-1)$,
+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
-$I=\int_{\mathbb R}V(s)\rho(s)\,ds$.
+$R_s(z)=-1/[sin(z-L)sin(z+L)]$, $R_c(z)=cos^2z$, put
+$I=int_{mathbb R}bbRV(s)rho(s),ds$.
[Homogeneous gas upper bound]
-As $m\to∞$, locally uniformly in $|\Im r|<.54\pi$:
+As $mto∞$, locally uniformly in $|Im r|<.54pi$:
-\log|P_m(e^r)|\le m^2I+(\frac38+(2)/(3\pi^2)\Re r^2)\log m+o(\log m).
+log|P_m(e^r)|le m^2I+(frac38+(2)/(3pi^2)Re r^2)log m+o(log m).
@@ -1655,74 +1655,74 @@ 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 $q_d\in\{\pm1\}$, a species, a contour, and a real position $s_d$.
+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
-t_{gh}(w)=\frac{R_{gh}(iw)}{R_{gh}(iw+L)R_{gh}(iw-L)},
+t_{gh}(w)=frac{R_{gh}(iw)}{R_{gh}(iw+L)R_{gh}(iw-L)},
-with measure $\tau dz/(2\pi i)$ and $\tau=2\sin(2L)/\sqrt C$.
+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 $S_T(w)=2sinh w\prod_{n\ge1}(1-e^{-2nT+2w})
+$frac12S_T(w+ijL)$ with $S_T(w)=2sinh wprod_{nge1}(1-e^{-2nT+2w})
(1-e^{-2nT-2w})$, and multiplying the integrand by
-$\sum_le^{-2Tl^2+4l\sum q_jw_j}$) we obtain a $T$-periodic
+$sum_le^{-2Tl^2+4lsum q_jw_j}$) 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 & $S$ & $D$ & $I_+'$ & $I_-'$ & $F$ \\
-$w_i$ & $1$ & $1$ & $-1$ & $1$ & $2$ \\
-$12\delta_i$ & $6$ & $1$ & $5$ & $-3$ & $-1$ \\
-$\sigma_i$ & $0$ & $1$ & $1$ & $-1$ & $0$
+$i$ & $0$ & $1$ & $2$ & $3$ & $4$
+hline
+label & $S$ & $D$ & $I_+'$ & $I_-'$ & $F$
+$w_i$ & $1$ & $1$ & $-1$ & $1$ & $2$
+$12delta_i$ & $6$ & $1$ & $5$ & $-3$ & $-1$
+$sigma_i$ & $0$ & $1$ & $1$ & $-1$ & $0$
The neutral-winding constraint on total particle counts is
-\sum_{g\in\{c,-\}}\sum_{i=1}^5w_i\nu_{g,i}=0.
+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
-Z^{(c)}_{ij}(p)=(2\pi)/(p)e^{(\delta_i-\delta_j)Lp}F^{(c)}_{ij}(e^{Lp}),
+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_j\frac{z^{[a+j]_8}}{z^8-1}+(1-\sigma)/(2)h_{[-a]_8},
-\quad h=(2,-1,1,-2c,2c,-2c,1,-1).
+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).
-Let $H_*=H(0;0)$, $A_0=\sum_{a\in(2)}H(a;1)$, $A_1=\sum_{a\in(2,3)}H(a;-1)$,
-$A_2=1$, $A_3=-1$, $A_4=A_1+A_3$. For $0\le i,j<4$, 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}_{\{i,j\}=\{1,3\}},
+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}},
-$F_{i4}=F_{i1}+F_{i3}+{\mathbf 1}_{i=3}-{\mathbf 1}_{i=1}$,
+$F_{i4}=F_{i1}+F_{i3}+{mathbf 1}bf1_{i=3}-{mathbf 1}bf1_{i=1}$,
$F_{4i}(z)=-F_{i4}(z^{-1})$, $F_{44}=F_{11}+F_{13}+F_{31}+F_{33}$,
where $P_{ij}$ collects lower-derivative residue rules
-$P_{ij}(z)=\sum_{a\in b_i,k\in b_j}H(a-k;sgn(j-i))$ for $i,j<2$
-and $sgn(j-i)$ for $i,j\ge2$, and $0$ otherwise.
+$P_{ij}(z)=sum_{ain b_i,kin b_j}H(a-k;sgn(j-i))$ 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 $\widehat\rho(p)=1/(2cosh Lp-1)$; then
+Let $K=K_{RR}$ and $widehatrho(p)=1/(2cosh Lp-1)$; then
-K=(1-e(p))K_0,\quad
-e(p)=(sinh(3Lp)-sinh(2Lp))/(sinh(4Lp))\in(0,1),
+K=(1-e(p))K_0,quad
+e(p)=(sinh(3Lp)-sinh(2Lp))/(sinh(4Lp))in(0,1),
-K_0=(2\pi)/(p)\coth(bp)&-csch(bp)\\
--csch(bp)&\coth(bp),\quad V_R=K{\mathbf 1}\widehat\rho.
+K_0=(2pi)/(p)coth(bp)&-csch(bp)
+-csch(bp)&coth(bp),quad V_R=K{mathbf 1}bf1widehatrho.
(Here $V_R$ denotes the potential contribution.)
@@ -1731,56 +1731,56 @@ Coercivity.:
The key analytic input is a positive-definite bound on $Z$.
[Coercivity]
-Put $h_*=3\pi b/4$ and $\alpha=2\pi^2/h_*$, and let $W=(w,w)^t$ be the
-weight column. Then for every real $p\ne0$ 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(\eta\alpha)/(p^2)WW^t+ε_0(1+p^2)^{-1}{\bf I},
+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 $Z_s=(Z+Z^t)/2$.
Proof. [Sketch]
-On each $R$-line, $K=(1-e(p))K_0$ with $e(p)\in(0,1)$ and $K_0$
-positive definite for $p\ne0$. The leading behavior at small $p$ is
-$K_0\sim2\pi/(p)(\coth bp{\bf I}-cschbp\,P)$ where
+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
+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(p\to0),
-\quad M_* real skew.
+Z(p)=alpha WW^t/p^2+M_*/p+O(1)quad(pto0),
+quad M_* real skew.
-Z(p)=(2\pi)/(|p|){\mathbf I}+2ff^t+O(e^{-ε|p|})\quad(|p|\to∞),\quad f={\mathbf e}_4.
+Z(p)=(2pi)/(|p|){mathbf I}bfI+2ff^t+O(e^{-ε|p|})quad(|p|to∞),quad f={mathbf e}bfe_4.
The grand sum.:
[Grand sum]
-For every integer $m\ge1$ and every sufficiently large real $T$,
+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^{c\cdot 2\pi iTr\,\sigma^t\nu/(4h_*)-S_{\rm bulk}(T)-S_{\rm gas}[\nu;r]},
+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 $S_{\rm bulk}(T)=(1)/(2T)\sum_{p\ne0}\widehat J^tK(p)\widehat J\,\widehat\rho(p)^2
--(1)/(2)(T/3-2\log2)(M_A-M_B)^2$,
+item $S_{rm bulk}(T)=(1)/(2T)sum_{pne0}widehatwidehat J^tK(p)J^tK(p)widehatwidehat JJ,widehatrho(p)^2
+-(1)/(2)(T/3-2log2)(M_A-M_B)^2$,
a real quadratic form;
-\item $S_{\rm gas}[\nu;r]=-(1)/(2T)\sum_{p\ne0}\sum_{d,e}Z_{g_de_d}(p)e^{-ip(s_d-s_e)}-(1)/(T)\sum_{linear}...$,
+item $S_{rm gas}[nu;r]=-(1)/(2T)sum_{pne0}sum_{d,e}Z_{g_de_d}(p)e^{-ip(s_d-s_e)}-(1)/(T)sum_{linear}...$,
the pairwise and single-particle interactions against real positions
-$s_d\in(-T/2,T/2)$;
-\item $ε_\nu\in\{\pm1\}$ is a fixed phase depending only on the multiplicity pattern;
-\item $c$ is a fixed scalar.
+$s_din(-T/2,T/2)$;
+item $ε_nuin{pm1}$ is a fixed phase depending only on the multiplicity pattern;
+item $c$ is a fixed scalar.
The sum and integral converge absolutely, and the sum localizes on
-\max_d|\Re s_d|\lesssim\log m+const.
+max_d|Re s_d|lesssimlog m+const.
@@ -1789,10 +1789,10 @@ Convergence and localization.:
Proof. [Sketch]
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
+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).
@@ -1802,8 +1802,8 @@ Consequently the whole sum is dominated by the localized region
Saturation at the two physical parameters.:
-We have the upper bound $\log|P_m(e^r)|\le m^2I+(3/8+2\Re(r^2)/(3\pi^2))
-\log m+o(\log m)$ for $|\Im r|<.54\pi$ (Proposition~prop:gas-upper),
+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.
@@ -1811,12 +1811,12 @@ The saturation statement.:
[Saturation at $y=1$ and $y=i$]
-\log P_m(1)=Im^2+\frac38\log m+o(\log m),\qquad
-\log P_m(i)=Im^2+\frac5{24}\log m+o(\log m).
+log P_m(1)=Im^2+frac38log m+o(log m),qquad
+log P_m(i)=Im^2+frac5{24}log m+o(log m).
Equivalently, the nesting ratio satisfies
-\mathcal Z_{m,m}(1)=(P_m(1))/(P_m(i))=m^{1/6+o(1)}.
+mathcal Z_calZ_{m,m}(1)=(P_m(1))/(P_m(i))=m^{1/6+o(1)}.
@@ -1824,81 +1824,81 @@ The reference determinant at $y=i$.:
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
-$h_0(d)=-\lambda\tan(\lambda d)/2$, $\lambda=\pi/(2b)=4/3$. 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^4\sin^8(b)/D(0)\big]^{m^2}=e^{Im^2}.
+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
-$X_j=e^{2i\lambda v_j}$ equals $\prod_{i<j}(X_i-X_j)^2/
-\prod(X_i+X_j)^2$ (clearing denominators, both sides alternate 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 $B(u)\tan(\lambda u)\to-\lambda\sin^2b$ at coincidence while
-$D(0)=C\sin^4b$.
+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 $y=i$.:
-Write $P_m(i)/P_0^{(\lambda)}$ as the product over $m$ "duplicate
+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)),
+prod_{k=1}^m(det(1+Delta_k))/(det(1)),
-where $\Delta_k$ is the rank-one modification between the $h$ and $h_0$
+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 $A$
+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 $j\in A$ to imaginary
+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
-\big|\det(1+\Delta_k)\big|\le e^{-ε\,dist(A_k,
+big|det(1+Delta_k)big|le e^{-ε,dist(A_k,
∂chamber)}.
Summing chambers and applying maximum modulus:
-\log\big|P_m(i)/P_0^{(\lambda)}\big|\le\frac5{24}\log m+O(1).
+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).
+log P_m(i)=Im^2+frac5{24}log m+o(log m).
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+(2\Re(r^2))/(3\pi^2)).
+f_m(r)=(log|P_m(e^r)|-Im^2)/(log m)
+-(frac38+(2Re(r^2))/(3pi^2)).
-From Proposition~prop:gas-upper, $f_m\le o(1)$ locally
-uniformly in $|\Im r|<.54\pi$. From (eq:P-m-i-final) at
-$r=i\pi/2$ (i.e. $y=i$), $f_m(i\pi/2)\to0$.
+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 $r=i\pi/2$ and
-reaching any compact subset of $|\Im r|<.54\pi$: the two ends of the
-chain give $f_m\to0$ in local $L^1$ throughout the strip. In
+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+\frac38\log m+o(\log m).
+log P_m(1)=Im^2+frac38log m+o(log m).
Combining (eq:P-m-i-final) and (eq:saturation-at-1):
-\mathcal Z_{m,m}(1)=P_m(1)/P_m(i)=m^{1/6+o(1)},
+mathcal Z_calZ_{m,m}(1)=P_m(1)/P_m(i)=m^{1/6+o(1)},
as claimed.
@@ -1908,7 +1908,7 @@ as claimed.
Crossings, cylinder barriers, and positive nesting.:
Section~c6:strip-asymptotics gave the strip-mass estimate
-$B_k\asymp k^{-1/4}$. 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.
@@ -1918,8 +1918,8 @@ 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 $x<\kappa$
+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 $x$.
@@ -1930,55 +1930,55 @@ Boundary crossings.:
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
+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 $K_l\asymp l^{-5/4}$.
+item The half-plane mass $K_l$ from a base port to another at lattice
+distance $l$ is decreasing in $l$ and satisfies $K_lasymp l^{-5/4}$.
Proof. [Sketch]
Contour comparison. The signed exit identity (Section~c6:strip-flux
-with $\beta=3/8$) gives
+with $beta=3/8$) gives
-\sum_{\gamma:p\to∂ D}\kappa^{\ell(\gamma)}
-\cos(\beta W(\gamma))=1,\quad c\le\cos(\beta W(\gamma))\le1.
+sum_{gamma:pto∂ D}kappa^{ell(gamma)}
+cos(beta W(gamma))=1,quad clecos(beta W(gamma))le1.
-Restricting to a convex subdomain and using $c=\cos(\beta\pi)>0$ 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
lower bound from the base-return identity
-$2\cos(\beta\pi/3)S=1-cA_T$ 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
-(K_{P-a}-K_{Q-a})\sin(\beta\pi)=D_1\sin(2\beta\pi)\ge0,
+(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 $\asymp l$: $K_l\lesssim l^{-5/4}$.
+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 $q=n_0\cdot(p_1-a)$,
+outward normal $n_1$, at depth $q=n_0cdot(p_1-a)$,
-\mathop{average}_{t\in[\alpha R,AR]}\sum_{q\ge\delta R}k(t,q)
-\ge c_{\alpha,A,\delta}R^{-1/4},
+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 $q\ge\delta R$ and
-$X\ge\eta R$, whence a second-moment bound
+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},
+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
+$mu[Y>0]gtrsim R^{-1/4}$ and the matching lower bound for $K_l$ by
monotonicity.
@@ -1987,7 +1987,7 @@ 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 $M\ge1$ with $H/M\ge R_0$,
+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
@@ -1997,8 +1997,8 @@ 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
+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
@@ -2012,16 +2012,16 @@ 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_{\gamma\subset S,\ \gamma essential simple}
-\kappa^{\ell(\gamma)}\ge c.
+sum_{gammasubset S, gamma essential simple}
+kappa^{ell(gamma)}ge c.
Proof. [Sketch]
Build confined arches via collar decompositions at two ports
-$p,q$ separated by $D\in[N/2-ε N,N/2+ε N]$. 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
+$S(t)$ 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
@@ -2032,60 +2032,60 @@ 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 $\mathcal P$ be
-its set of simple polygons. For $0\le\beta'\le3/8$, put
-$\alpha=\pi\beta'/3$, $x=(2\cos\alpha)^{-1}$, and
+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
-\omega(L)=x^{|L|}\cdot
--2\cos(4\alpha),&L contractible,\\
--2\cos(2\alpha),&L essential.
+omega(L)=x^{|L|}cdot
+-2cos(4alpha),&L contractible,
+-2cos(2alpha),&L essential.
-For $\mathcal A⊆\mathcal P$:
+For $mathcal A⊆calA⊆mathcal P$:calP$:
-Q(\mathcal A)=\sum_{\substack{\mathcal F⊆\mathcal A\\
-polygons in \mathcal F disjoint}}
-\prod_{L\in\mathcal F}\omega(L).
+Q(mathcal A)=calA)=sum_{substack{mathcal F⊆calF⊆mathcal AcalA
+polygons in mathcal FcalF disjoint}}
+prod_{Linmathcal F}calFomega(L).
[Positivity]
-For every $N\ge1$, $h\ge1$, $0\le\beta'\le3/8$, $\mathcal A⊆\mathcal P$:
-$Q(\mathcal A)>0$.
+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 $N,h\ge2$: elementary polygons (six-centre loops around
+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
-Q(\mathcal P)=\sum_{\gamma:p\to∂ D}x^{|\gamma|}
-e^{i\beta'W(\gamma)}Q(\{L:L\cap\gamma=∅\}).
+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 $\mathcal A$ must be all of
-$\mathcal P$ 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 $mathcal A$calA$ must be all of
+$mathcal P$calP$ 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 $-t\kappa^{|L|}$ with $t\in[0,\sqrt2]$. 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)\ge\exp\Big(t\!\sum_{\ell\ essential,\ \ell\cap\gamma\ne∅}\!\kappa^{|\ell|}\Big).
+H_H(gamma;t)geexpBig(t!sum_{ell essential, ellcapgammane∅}!kappa^{|ell|}Big).
Combining (eq:cylinder-deletion-ratio) with
(eq:slab-essential-mass) and (eq:strip-upper) gives
--\log Z_H(\sqrt2)\le CH/N\qquad(H\ge N),
+-log Z_H(sqrt2)le CH/Nqquad(Hge N),
and hence
-\int_1^{\sqrt2}\sum_{\ell essential}\kappa^{|\ell|}
-H_H(\ell;t)\,dt\le CH/N.
+int_1^{sqrt2}sum_{ell essential}kappa^{|ell|}
+H_H(ell;t),dtle CH/N.
Cylinder barriers and height tails.:
@@ -2093,11 +2093,11 @@ 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
+item 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
+item From a fixed bottom port in a semi-infinite cylinder, arches
reaching height $s$ have mass at most $Ce^{-cs/N}$.
-\item For $s\ge1$, all polygons meeting a fixed level with height span
+item For $sge1$, all polygons meeting a fixed level with height span
in $[s,2s]$ have total mass at most $poly(N,s)
e^{-cs/N}$.
@@ -2107,7 +2107,7 @@ 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 $(jN,(j+1)N)$.
+$H_H(gamma;t)gtrsim e^{c's/N}$. 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.
@@ -2130,8 +2130,8 @@ 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
-$\mathcal Z\lesssim Z(CN^{1+2\xi})^2$ 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).
@@ -2141,14 +2141,14 @@ 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)
+U_D(e)asymp Z_D(y)+Z_D(z)
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
+$S_e=Z_e+(M_y+M_z)/sqrt2$, 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
@@ -2163,7 +2163,7 @@ The saturation bound of Section~c6:saturation identifies a nonzero
normalization at exponent level. We now sharpen two inputs to bounded
factors: the geometric comparison between plane and cylinder, and the
analytic estimate of the normalized edge contributions. Both are used
-in the sewing argument (\Ssec:sewing).
+in the sewing argument (Ssec:sewing).
Deletion ratio and the first-hit comparison.:
@@ -2171,7 +2171,7 @@ Let $D$ be either a finite cylinder between flat caps, or a finite
convex tiled polygon in the plane. Write
Q_D=
-Z_H(\sqrt2),&D cylinder,\\
+Z_H(sqrt2),&D cylinder,
1,&D plane,
@@ -2183,80 +2183,80 @@ centres:
[Deletion monotonicity]
$Θ_D(K)$ is increasing in both the set of available polygons and
-the deleted set $K$, for $0\le t\le\sqrt2$.
+the deleted set $K$, for $0le tlesqrt2$.
Proof.
For a single deleted polygon $v$:
-Q_P/Q_{P∖ v}=1-t\kappa^{|v|}\cdot Q_{P∖ N[v]}/Q_{P∖ 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
+(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 $L\subset D$ and a boundary source $a$, let
+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:
-1\ge P(a,L)\ge
-\sum_{\substack{\gamma:a\to∂ D\\ \gamma\cap L\ne∅}}
-\kappa^{|\gamma|}\cos(\beta W(\gamma))H_D(\gamma)\ge0.
+1ge P(a,L)ge
+sum_{substack{gamma:ato∂ D gammacap Lne∅}}
+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=\Im\sum_{a\in∂ D}\sum_{\gamma:a\to∂ D}
-cr_\Gamma(\gamma)\,\kappa^{|\gamma|}e^{i\beta W(\gamma)}H_D(\gamma).
+I=Imsum_{ain∂ D}sum_{gamma:ato∂ D}
+cr_Gamma(gamma),kappa^{|gamma|}e^{ibeta W(gamma)}H_D(gamma).
Then
-2\!\!\sum_{L contractible enclosing y}\!\!\kappa^{|L|}H_D(L)\sum_aP(a,L)
-\le|I|+\sqrt2\!\!\sum_{L essential}\!\!\kappa^{|L|}H_D(L)\sum_aP(a,L).
+2!!sum_{L contractible enclosing y}!!kappa^{|L|}H_D(L)sum_aP(a,L)
+le|I|+sqrt2!!sum_{L essential}!!kappa^{|L|}H_D(L)sum_aP(a,L).
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: $J=-(\mathcal Y(y)/Q_D-1)/\sqrt2$.
+(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 $|E|\le
-\sqrt2\sum_{essential}\kappa^{|L|}H_D(L)\sum_aP(a,L)$ (turn
-increments $\pm4\pi/3$ versus $\pm2\pi/3$). 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 $s\ge1$ 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.
+sum_{substack{L contractible enclosing y
+diam(L)in[s,2s]}}kappa^{|L|}le C.
Proof. [Sketch]
-Take a regular lattice hexagon of inradius $\asymp s$ around $y$. Only
+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.
+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
+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.
@@ -2270,38 +2270,38 @@ independently of $N$.
Proof. [Sketch]
-Good heights. Put $F_H(t)=-\log Z_H(t)$. Superadditivity of
+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},qquad 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
+$F_{H_j}(sqrt2)-kF_{H_j/k}(sqrt2)le uH_j/(4N)$, and then
-\sum_{L essential in H_j-slab}\kappa^{|L|}H_{slab}(L)
-\le e^{uH_j/N}.
+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 $H$,
-\sum_{\substack{L contractible enclosing y\\
-span_{vert}\in[H/4,H/2]}}\kappa^{|L|}\le Ce^{-cH/N},
+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}
+(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\\
-lift-diam>c_2N}}\kappa^{|L|}\le C_{c_2}.
+sum_{substack{L contractible enclosing y
+lift-diam>c_2N}}kappa^{|L|}le C_{c_2}.
The remaining finite collection of exceptional bands is bounded by the
planar annular bound (eq:annular-mass) applied to the corresponding
@@ -2311,36 +2311,36 @@ diameter intervals.
Classwise convergence of the normalization.:
[Classwise convergence]
-For $r\in\{0,i\pi/2\}$ put $\sigma(r)=3/8+2r^2/(3\pi^2)$ and
-$\mathcal A_m(r)=e^{-Im^2}m^{-\sigma(r)}P_m(e^r)$. There exist a fixed
-integer $Q\ge1$, a $\delta>0$, and positive constants
-$\mathfrak a_b(r)$, $0\le b<Q$, 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)=\mathfrak a_{m\bmod Q}(r)+O(m^{-\delta}).
+mathcal A_m(r)=calA_m(r)=mathfrak a_fraka_{mbmod Q}(r)+O(m^{-delta}).
Consequently
-\mathcal Z_{m,m}(1)=P_m(1)/P_m(i)\asymp m^{1/6}.
+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
-$T=2a_0^{-1}\log m+g+i\theta$, $|\theta|\le\pi/2$, for a suitably large
+$T=2a_0^{-1}log m+g+itheta$, $|theta|lepi/2$, for a suitably large
fixed $g$. The stability estimate (eq:Z-coercive) and
localization (eq:gas-localization) continue to hold with the
-coercivity error $O(1/g)\mathcal Q$ absorbed by $g$ large; the period
+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.
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 $\mathcal A_m(r)$. We show
+$e^{-2Am^2e^{-2T}}e^{-Im^2}m^{-sigma(r)}$ 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.
+$pi T/(16L)=(5)/(24)log m$ 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
@@ -2349,10 +2349,10 @@ agreement of the two retained integrands up to $e^{-cT'}$.
Discrete phases. The $m$-dependent phase factors are roots of
unity from fixed logarithmic branch multiplies; the entire phase
-depends only on $m\bmod Q$ 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).
@@ -2372,47 +2372,47 @@ 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 $k\ge2$, and
-$N=kd$. The polygon weight is $\kappa^{\#triangle centres visited}$.
+$d=P_1+P_2$ 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
intervals between the marks even in staircase steps.
Finite marked contraction.:
-Fix integers $k\ge1$, $0\le l,h\le k$, and $m,n\ge0$. Take two lists
+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 $\delta\ne0$, the contraction
+For $deltane0$, the contraction
-Y=\big<\Psi_b\big|\,\delta^{\sum_A p_j}\big|\Psi_t\big>,
+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
-Y=Q_{lh}\,\mathcal J_{\rm ext}\!\!
-\sum_{\substack{a_j\in\{-1,0,1\}\\
-\sum_Aa_j-\sum_Ba_j=h-l}}
-(i/\delta)^{\sum_Aa_j}\,\mathcal W(a)\,\mathcal M(a),
+Y=Q_{lh},mathcal J_calJ_{rm ext}!!
+sum_{substack{a_jin{-1,0,1}
+sum_Aa_j-sum_Ba_j=h-l}}
+(i/delta)^{sum_Aa_j},mathcal W(a)calW(a),mathcal M(a),calM(a),
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 $\mathcal J_{\rm ext}$ is a
+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 $\mathcal W(a)=\prod_jc_{a_j}\prod_{j<j'}P_{a_ja_{j'}}(v_{j'}-v_j)
-\prod_{j<j',\,same list}D(v_{j'}-v_j)$, with $P$, $D$ from
+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
Section~c6:periodic-polynomial.
-\item $M^1_+(u)=(1,\cos(u+L),J(u))$, $M^1_0=1$ extracted by
+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 $\mathcal M(a)=\prod_{j\in A\cup B}\prod_{e\inextras}
-M^{d(j,e)}_{a_j}(x_e-v_j)$ with $d(j,e)\in\{1,2\}$ determined by list
+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
and spin type.
@@ -2432,16 +2432,16 @@ 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}
+frac{Y}{f_{m+n+2}(0)^2}
is the critical mass of individual simple polygons through the two
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 $E$, $H$
+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.
@@ -2451,28 +2451,28 @@ 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
sites, the two-edge marked mass satisfies
-K_2\le(m+1)^{-4/3+o(1)}.
+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
+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.
The gas estimate at this shifted potential gives the claimed bound.
The magnetic test potential.
-In the residue lift, using the periodized variable $w=i\log$, 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))
@@ -2492,8 +2492,8 @@ 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 $\asymp\log d$ and a full
-period of size $\asymp\log N$. All constants below are independent of
+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
@@ -2514,11 +2514,11 @@ bounded below positively. Indeed, writing the domain in lattice
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 $[-\pi/3,2\pi]$ or in $[0,2\pi+\pi/3]$, and
-$(3/8)(2\pi+\pi/3)<\pi$ gives the positivity. For the through-mass
-at height $r$, take a lattice hexagon of radius $\asymp r$ not
+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
@@ -2530,10 +2530,10 @@ 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, $0<\lambda\le1/2$. After a rapidity translation, put
+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
-$r_0=1/4+\lambda/2$.
+$r_0=1/4+lambda/2$.
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
@@ -2543,19 +2543,19 @@ 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)).
+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 $u$.
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
+$sum_e q_e i a_e$ 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
in rapidities.
@@ -2569,85 +2569,85 @@ 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 $E(p)=\sum_e e^{-p a_e}{\bf e}_{g_e}$.
+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 (${\cal M}_{R,E}n_E=-K\widehat\rho E$)
+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 $4q_eq_f/p^2+4q_eq_f(a_e-a_f)/p$
with zero finite part. Then:
-\item The bulk log term is
+item The bulk log term is
-S(T)=\frac1{2T}\sum_p E(-p)^tK(p)E(p)\widehat\rho(p)^2
--\tfrac12(T/3-2\log2)(M_A-M_B)^2.
+S(T)=frac1{2T}sum_p E(-p)^tK(p)E(p)widehatrho(p)^2
+-tfrac12(T/3-2log2)(M_A-M_B)^2.
Determinants and measure conventions of (eq:legacy-P-star) are
unchanged.
-\item The chemical potentials per final particle (before the magnetic
+item The chemical potentials per final particle (before the magnetic
test) are
-2\pi i\,\sigma_i\!\!\!\sum_{e:\,g_e=g_i}\!\!\!
-\int_0^{s+i(v_e+\delta_i L)}\!\!\!
-\sum_{\nu\in\mathbb Z}\rho(z+\nu T)\,dz,\quad
-\sigma=(0,1,1,-1,0).
+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
+sigma=(0,1,1,-1,0).
-Fugacity still brings $Tr_j^2/(4h_*)-r_j(u_*+\bar L)$.
+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 $q^t\gamma'_{\rm raw}
-=A_L+\sum_e q_ea_e+\pi l/2$. The extended matrix has the same
-charge/offset/winding poles, and the saddle term $\widehat\rho E$
+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
+$sum_e q_ea_e$ 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
-$-2\pi\sigma_i e^{(v_e+\delta_i L)p}\widehat\rho(p)/p$; 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
+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
-\[
-{\bf1}^t(J(p)-K(p)\ell)=2\pi W^t/p-2h_*c^t+O(p),
-\]
-so $u_*+\bar L=c^tn-(\pi i/h_*)\sum W_ds_d$. (Here $J$ is the
+[
+{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
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
-M_*=-2\pi(\cal Wtilde+Wc^t-cW^t),
+Z=alpha WW^t/p^2+M_*/p+O(1),qquad
+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 $-2\pi\cal Wtilde$ 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 $-2\pi ijc^tn$, 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_d\eta_{i_d}\exp[-2\pi i c_{i_d}H_m(s_d)]
+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
+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
@@ -2657,12 +2657,12 @@ 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 $\Re s>0$,
+regular contact self term carries half weight. For $Re s>0$,
-G_{ab}(s)=\sum_{l\ge1}\sum_{k=0}^{r_*}C_{ab,l,k}s^ke^{-\omega_ls},
+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
down in imaginary periods yields the residues above the separated
@@ -2680,36 +2680,36 @@ 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
+$a_0'=pm6Li$, 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.
+-h_*^{-1}int(pi H+ia_0')^2,ds+2a_0'c^tn.
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 $x\to y$ require outgoing height of $x$ to equal incoming of
+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
+$pi^2/h_*int H^2$) 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 $S_0$,
-\sum A(x_i,x_{i+1})\ge4ε_S\sum|x_i|_0^2-C_Sn_{\rm cells},
-\qquad|x|_0^2=N_x^2+H_{x,\rm in}^2,
+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,
$ε_S>0$ at least inverse-polynomial in $S_0$. There is a Borel
-$\phi$ on states away from coincidences with
-\[
-A(x,y)\ge2ε_S(|x|_0^2+|y|_0^2)-C_S+\phi(y)-\phi(x),
-\]
+$phi$ 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
both endpoints, including for small-slope complex cell deformations.
@@ -2717,20 +2717,20 @@ 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
+$D_d$ 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 $x_i\to x_{i+1}$ is
-\[
-\exp a^\dagger(U)\ \Gamma(D_{\ell'})\ \exp a(V),
-\]
-with $V=\sum_{b\in x_{i+1}}v_bD_{d_b^L}$,
-$U=\sum_{a\in x_i}D_{\ell'+d_a^R}u_a$, symmetric tensor powers, and
+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
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
+$|U|+|V|le Ce^{-ε S_0}(N_x+N_y)$; 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
@@ -2738,51 +2738,51 @@ energies.
[Homogeneous interval operators]
On the ambient cell space with the Fock fibers above, let
-$\mathcal T_\pm$ be homogeneous cell operators with all five color-$A$
+$mathcal T_calT_pm$ 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
+(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
-$spr(\mathcal T_\pm)<\exp(S_0/2)$.
+$spr(mathcal T_calT_pm)<exp(S_0/2)$.
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
+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
+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, $spr(\mathcal T)=limsup_m\|\mathcal T^m\|^{1/m}$;
+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 $m_0S_0$, $m_0\to∞$.
+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 $M_A=0,M_B\to∞$ even at zero. The original sum has no
+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
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
+$int(H-j)=sum W_ds_d$, 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 $K_{0,BB}=(2\pi/p)\coth(bp)$, and dressing
+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
+$prod_{p>0}(1-e^{-2bp})/(1+e^{-2bp})$ 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
@@ -2790,53 +2790,53 @@ polynomially in $T$.
Real same-color kernel. For positivity put
$D_0(z)=(z^2-z+1)(z^4-z^2+1)$, $t=e^{x/12}$,
-$H_{ij}=t^{-36+12(\delta_i-\delta_j)}D_0(t^{12})F_{ij}^{(0)}(t^{12})$
+$H_{ij}=t^{-36+12(delta_i-delta_j)}D_0(t^{12})F_{ij}^{(0)}(t^{12})$
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 $t^j$ ($1\le j\le36$) enters with
-multiplier $sinh(\beta x)/sinh x$, $\beta=j/12$. With
+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
-\[
-u_0=\beta,\quad u_1=\beta(\beta^2-1)/6,\quad
-u_2=\beta(\beta^2-1)(\beta^2-4)/120,
-\]
+[
+u_0=beta,quad u_1=beta(beta^2-1)/6,quad
+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
-\[
-U_2=\sum_ju_2B_j-\tfrac12\sum_{12\nmid j}
-(u_2B_j+|u_2|diag(|B_j|_{\rm ent}{\bf1})).
-\]
+$U_k=sum_ju_kB_j$ ($k=0,1$) 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 $5\times5$ operator norm at most $5/2$, so the exact matrices are
+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 $2\pi/p$, on the diagonal are
-$(sinh3x+sinh x\mpsinh2x)/cosh3x$ and off-diagonal
+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 $cosh(hp)/cosh(bp)$, $0\le h<b$). The regular
-diagonal is at least $2\log L$ (each $F^{(0)}$ diagonal is at least
+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
+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
+[
+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
either sign of the tilt.
For later use, the spectral-radius formula gives a fixed block length
$q$ and $ε>0$ with
-$\|\mathcal T_\pm^q\|\le\exp((1/2-ε)qS_0)$ for both tilts.
+$|mathcal T_calT_pm^q|leexp((1/2-ε)qS_0)$ 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.
@@ -2848,72 +2848,72 @@ 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,
-as in the balanced case (\Sc6:subheading-16). On the right,
-the mean $\bar a$ gives $r=-2\bar a$ and scalar $(M_A+M_B)\bar a$; on
+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 $a-\bar a$. Completing Cartan nonzero modes as in the
+right insert $a-barbar a$.a$. Completing Cartan nonzero modes as in the
balanced case yields an extra external scalar
-S_a(T)=\frac1T\sum_p\widehat\rho(p){\bf1}^tE(-p)\widehat a(p)
-+\frac1T\sum_{p\ne0}(Q_+(p)-1/h_*)\widehat a(-p)\widehat a(p),
+S_a(T)=frac1Tsum_pwidehatrho(p){bf1}^tE(-p)widehatwidehat a(p)a(p)
++frac1Tsum_{pne0}(Q_+(p)-1/h_*)widehatwidehat a(-p)a(-p)widehatwidehat a(p),a(p),
and the local height-square becomes
--h_*^{-1}\int(\pi H+ia(s))^2\,ds.
+-h_*^{-1}int(pi H+ia(s))^2,ds.
The added particle-source nonzero-mode multiplier is
-$-J(-p)^t{\bf1}Q_+(p)$; its simple pole $+2\pi W/(ph_*)$ acting on
-$a-\bar a$ gives the mean-zero primitive of $-i(a-\bar a)$, and
+$-J(-p)^t{bf1}Q_+(p)$; 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
(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
-(2c_i-\ell_{Σ i})a(s)
-+\!\!\!\!\sum_{orig.\ left member}\!\!\!\!\log U_T(s+i(\delta_i+b_{i'})L)
+(2c_i-ell_{Σ i})a(s)
++!!!!sum_{orig. left member}!!!!log U_T(s+i(delta_i+b_{i'})L)
+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
-$[-J(-p)^t{\bf1}Q_+(p)-2\pi W/(ph_*)-(2c-\ell_Σ)]_i/(-ip)$.
-Here $Q_+=1/h_*+O(p^2)$ fixes the constant $2c-\ell_Σ$, whose
+$[-J(-p)^t{bf1}Q_+(p)-2pi W/(ph_*)-(2c-ell_Σ)]_i/(-ip)$.
+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 $g\gg S_0$ large enough to meet
+$q$, 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
-$d,N,\lambda$.
+$d,N,lambda$.
-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
-$s=t+iy(t)+(T-R)\chi(t)$, $-R/2\le t\le R/2$, with $y$ periodic,
+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
+$-theta_0L$ 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
+$chi(pm R/2)=pm1/2$, 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 $\pm t>0$
+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
direct multipliers in (eq:legacy-P-33) tend to these phases
-modulo $2\pi i$ 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)
+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
+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.
+outside $|t|le a_0^{-1}log d$ and away from the period edges.
Proof of 1. The product formula and derivative periodizations
@@ -2922,48 +2922,48 @@ 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 $i\delta_iL$
+$I_-'$, 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
(eq:legacy-P-P3), gives all asserted bounds. Widths on the
unshifted ratios are bounded away from singularities since
-$-3/4\le x_e/L\le1/2$.
+$-3/4le x_e/Lle1/2$.
Proof of 2. On the basic line the imaginary part of a single
primitive at argument $t+iLY$ is
-\[
-B(u,Y)=\frac1{2\pi}atanh
-(4\sqrt3cosh u\sin(\pi Y/3))/(3+4sinh^2u+4\sin^2(\pi Y/3)),
-\quad u=a_0t,
-\]
+[
+B(u,Y)=frac1{2pi}atanh
+(4sqrt3cosh usin(pi Y/3))/(3+4sinh^2u+4sin^2(pi Y/3)),
+quad 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 $\ge ce^{-|u|}$
-($Y\ge 2/3$ 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
-\[
-(2\pi B)/(4\sqrt3cosh u\sin(\pi Y/3)/(3+4sinh^2u))
-\]
-between $1$ and $atanh(\sqrt z)/\sqrt z$, with
-$z=4\sin^2(\pi Y/3)/(3+4sinh^2u)$. For $D$ with $\lambda\le1/3$,
+[
+(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 $lambdale1/3$,
it suffices that
-\[
-(1-\lambda)\sin(\pi Y/3)-1.33\lambda\sin(\pi(1-Y)/3)>0;
-\]
-splitting at $\lambda=1/4$ gives $\ge17/600$ for
-$0\le\lambda\le1/4$ (via $\sin(\pi/9)>1/3,\sin(2\pi/9)<2/3$) and
-$\ge337/6000$ for $1/4\le\lambda\le1/3$ (via $\sin(11\pi/72)>9/20$
-and $\sin(13\pi/72)<11/20$), using
-$atanh(\sqrt z)/\sqrt z<133/100$ for $z\le5/9$ (from
+[
+(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 $I_-'$,
-$Y=\lambda/2\le1/4$, so $z<1/10$ and
-$atanh(\sqrt z)/\sqrt z<11/10$; then
-$(11/10)(1-\lambda)\sin(\pi\lambda/6)\le(11\pi/60)\lambda
-<\lambda\sin(\pi/4)\le\lambda\sin(\pi(1-Y)/3)$, giving the negative
-sign with margin $\lambda e^{-|u|}$. Bending improves these
-comparisons for small $\theta_0$; the primitive limits to $\pm1/2$
+$Y=lambda/2le1/4$, 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
@@ -2989,21 +2989,21 @@ equal to the original trigonometric sum at nome zero.
A bounded product through the core and tails.:
-Choose $u_0\ge0$ 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
+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 $12\pi L/h_*$. Terms from purely
+[
+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
imaginary tangents have unit modulus for this leading tilt. All
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
-($M_g\ge d$ and no species has $W=0$). We replace this real linear
+($M_gge d$ 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
@@ -3017,12 +3017,12 @@ 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
+$p_i(x)=delta,depth_iN_x$ 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
-\[
-K_i^{\rm sym}(x,y)=\exp\!\Bigl[\frac{p_i(x)-p_{i+1}(y)}2\Bigr]K_i(x,y).
-\]
+[
+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
telescope pointwise, including the interfaces where the exterior
@@ -3032,21 +3032,21 @@ 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
-$C\exp(-c\,depth)$, 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
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 $H\ne0$, and vanishes when $H=0$. 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
-$1+C\exp(-c\,depth)$ 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
@@ -3061,16 +3061,16 @@ 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
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 $q$
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
+$exp((1/2-ε)qS_0)$. 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
+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
@@ -3081,7 +3081,7 @@ the core scalar but excluding (eq:legacy-P-24),
(eq:legacy-P-31) and the determinant, the overall log modulus is
at most
--2R+\tfrac52(R-2u_0)+O(1).
+-2R+tfrac52(R-2u_0)+O(1).
@@ -3096,157 +3096,157 @@ multiplicities.
External scalars.: Uniformly on the circle,
(eq:legacy-P-24) equals
-\[
-\tfrac12\sum_{e,f}I_{g_eg_f}(v_e-v_f)+C_2e^{-2T}+O(1)
-\]
+[
+tfrac12sum_{e,f}I_{g_eg_f}(v_e-v_f)+C_2e^{-2T}+O(1)
+]
for a site-data constant $C_2$, where
-\[
-I_{gh}(v)=2q_gq_h\log2+\int_{\mathbb R}\!\!
-[K_{gh}(p)\widehat\rho(p)^2cosh(vp)-4q_gq_h/p^2]\,(dp)/(2\pi).
-\]
+[
+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 $O(N^2e^{-a_0\Re T})$,
-picking corrections at poles $\pm2i$, and at most a simple pole at
-$\pm ia_0$ in the color-difference channel. Then
+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)^2\tan^2(\lambda_bv)
-1/D(v),&g=h,\\1,&g\ne h,
-\quad B(v)=(\sin(v+b)\sin(v-b))/(\sin v).
+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).
Verification: for real $v$ the multiplier
-$(p/2\pi)K_{gh}\widehat\rho^2$ on $p>0$ is
+$(p/2pi)K_{gh}widehatrho^2$ 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
-$\sum_{k\ge1}c_ke^{-kLp}$ 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 $\sum_k-c_k\log(1-v^2/(kL)^2)$, matching
+$sum_{kge1}c_ke^{-kLp}$ 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
(eq:legacy-P-35) by the sine product; the constant follows by
-$v=iV$, $V\to+∞$, 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
-\[
-\sum_e\log(J(x_e-v_e))/(J(v_e-x_e))+C_1e^{-T}+O(1).
-\]
+[
+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 $a$
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 $O(e^{-2\Re T+4|\Re s|})$, whose
-convolution with each $\rho(s+iv_e)$ contributes the coefficient and
-costs $O(1+Ne^{-a_0\Re T/2})$. The principal per-background integral
+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
-\[
-2\pi i\widehat\rho(p)(z-z^7)/(1-z^8)
-=2\pi i(z^2+z^3-z^5-z^6)/(1-z^8),\quad z=e^{-Lp},
-\]
+[
+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},
+]
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
exponential analytic multiplier and use maximum modulus. The
-two-color transform determinant adds $(3/8)\log N+O(1)$. For the
-remaining determinant put $\mathfrak d(p)=-\log(1-e(p))$ and use
-\[
-2\sum_{n\ge1}\mathfrak d(2\pi n/T)=\sum_{n\in\mathbb Z}
-\mathfrak d(2\pi n/T)-\mathfrak d(0).
-\]
-Since $\mathfrak d$ is even, regular at zero, and exponentially
+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
decaying, the periodization contributes
-$\pi T/(16L)-\mathfrak d(0)+O(e^{-c\Re T})$ 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
-$h_0(v)=-\lambda_b\tan(\lambda_bv)/2$ instead of $h$ (also by
+$h_0(v)=-lambda_btan(lambda_bv)/2$ instead of $h$ (also by
Schur--Pfaffian at shifted variables).
The Pfaffian ratio.: We need
-\log(f_N^2/f_N^{(0)2})\ge(5/24)\log N-O(1).
+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
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 $R(p)=\nu/\nu_0\ge1$. 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
+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
-$g\nu_0$; confluent limits preserve the comparison.
+$gnu_0$; confluent limits preserve the comparison.
-Uniform $A\log N+O(1)$. The doubled linear statistic has the
-form $A\log N+O(1)$, with $A$ independent of the rapidities and of
+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 $C(v)=\prod_j[-i\cot(\lambda_b(v-v_j))]$. If $\Gamma$ positively
+and $C(v)=prod_j[-icot(lambda_b(v-v_j))]$. If $Gamma$ positively
encircles the sites and initially no other poles,
-\[
-2\Bigl.(d)/(dt)\log(\Pf H_t)/(\Pf H_0)|_{t=0}
-=-(4)/((2\pi i)^2)\int_\Gamma\!\int_\Gamma
-C(v)C(w)h_0(v-w)h_g(v-w)\,dv\,dw.
-\]
+[
+2Bigl.(d)/(dt)log(Pf H_t)/(Pf H_0)|_{t=0}
+=-(4)/((2pi i)^2)int_Gamma!int_Gamma
+C(v)C(w)h_0(v-w)h_g(v-w),dv,dw.
+]
Cancelling common Schur-minor factors gives the undoubled derivative
-$-4\sum_{i<j}r_ir_jh_0(v_i-v_j)h_g(v_i-v_j)$ with
+$-4sum_{i<j}r_ir_jh_0(v_i-v_j)h_g(v_i-v_j)$ with
$r_i=Res_{v=v_i}C$; the normalized double integral is
-$2\sum r_ir_jh_0h_g$ because diagonal residues vanish (both kernels
+$2sum r_ir_jh_0h_g$ 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 $g\nu_0$; its $1/p$ portion has bounded odd sine
+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 $w$
+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 $C(v)C(v\pm b)=1$, 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
+$exp(-cNe^{-a_0|Im v|})$, 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
-\[
-\int_0^{a_0^{-1}\log N}e^{-cNe^{-a_0t}}\,dt
-=a_0^{-1}\int_1^N e^{-cx}(dx)/(x)=O(1),
-\]
+[
+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 $w=v\pm b$ contributes a constant residue. This proves
-$A\log N+O(1)$ uniformly in the rapidities, including confluent
+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 $A\ge5/24$. 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 $A\ge5/24$, 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.
Assembly.: The log of the normalized marked-polygon mass
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
-\[
--\tfrac43\log N+(15)/(8)\log(N/d)+O(1).
-\]
+[
+-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
+[
+-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
@@ -3265,17 +3265,17 @@ two-mark estimate into a second length moment.
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
+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
+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
confined within that tolerance around the straight segment in
-direction $m$, with mass $\gtrsim s^{-1/4}$; the unrestricted mass 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
+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 $O(l^{-5/4})$.
+$gtrsim l^{-5/4}$; 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
@@ -3285,26 +3285,26 @@ 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 $s_1\in[as,2as]$; 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 $m$
+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
+$gtrsim (s^{-1/2})^2/s^{-3/4}=s^{-1/4}$. 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 $a\to Q$ back towards $P$ can only exit between
-$a,P$, with phase $z^{-1}$; conversely from $a\to P$ via the arc to
-$Q$, the residual chord has $|W|\le\pi$. After multiplication by
-$z=e^{i\beta\pi}$ the outgoing imaginary coefficients are therefore
+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
+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.
@@ -3314,9 +3314,9 @@ Two exterior sewing estimates.:
Take a triangular-lattice vertex as origin and pure-support domains
-$D=D(0;\mathbf b)$ with $|b_j-R|\leε R$ for a fixed
-$0<ε\le1/20$; 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.
@@ -3335,11 +3335,11 @@ 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
the conditional density bound becomes an atom bound
-$C\min(1,Kd_0/(s-r))$, sufficient for all averages below. Boundary
+$Cmin(1,Kd_0/(s-r))$, 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 $l\asymp|p-q|$ (at least constant order) and by
+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
@@ -3348,39 +3348,39 @@ deterministic event given $h,D_h$.
[Exterior sewing]
For the nested family, endpoint bin, and deterministic chord event
-just specified, the average $\overline E$ over the level and support
+just specified, the average $overlineoverline E$E$ 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
-$\overline E\le Cl^{3/4}\mathcal K$, where $\mathcal K$ is the
+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
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
+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
bridges between those caps, with base start free in a horizontal
interval of size $CR$ containing the required chord event for at
least one choice of the supports. Then
-$B^*\ge cR^{-1/2}(l/R)^{C_0}l^{1/2}\overline E$, for universal
+$B^*ge cR^{-1/2}(l/R)^{C_0}l^{1/2}overlineoverline E$,E$, for universal
$c,C_0>0$.
Proof.
The two completions are shown in Figure~fig:sewing.
-\input{figures/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
+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
+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 $\rho l$
+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 $\ge\rho l$ ($\rho>0$ fixed small). Each enlarged
+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.
@@ -3391,7 +3391,7 @@ 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
@@ -3414,11 +3414,11 @@ 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 $x,y\ge0$ 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
+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
@@ -3431,21 +3431,21 @@ Averaging intermediate levels. Fix one corridor construction
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
-$p\to q$ 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 $\mathbf u$, let
-$\mathbf1_{\mathbf u}(\eta)$ 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
-$Y(\eta)=\mathbb E_{\mathbf u}\mathbf1_{\mathbf u}(\eta)$ 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
-\[
-\int Y\,d\nu\ge
- c^kl^{-5/4}P,&case 1,\\
+[
+int Y,dnuge
+ c^kl^{-5/4}P,&case 1,
c^kP,&case 2.
-\]
+]
For two lists, split each branch at both choices of each varied
crossing, giving a difference bridge there. Nonincident pieces stay
@@ -3460,43 +3460,43 @@ $O(l^{-5/4})$ 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
-\[
-\int Y^2\,d\nu\le
- C^kl^{-5/4}P^2,&case 1,\\
+scale $asymp R$. Consequently
+[
+int Y^2,dnule
+ C^kl^{-5/4}P^2,&case 1,
C^kR^{1/2}P^2,&case 2,
-\]
+]
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
-\[
-\nu\{Y>0\}\ge((\int Y\,d\nu)^2)/(\int Y^2\,d\nu)\ge
- c_1^kl^{-5/4},&case 1,\\
+[
+nu{Y>0}ge((int Y,dnu)^2)/(int Y^2,dnu)ge
+ c_1^kl^{-5/4},&case 1,
c_1^kR^{-1/2},&case 2,
-\]
+]
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
-$c_1^k\ge c\min(1,(l/R)^{C_0})$ for a fixed $C_0>0$. These are the
+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 $F\asymp l^{-5/4}$ (case 1) or
-$F\asymp R^{-1/2}\min(1,(l/R)^{C_0})$ (case 2), pointwise for
+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
+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 $\le\mathcal K$ 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 $lemathcal K$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
chord-event indicator times the exterior trimming factor, so
-$0\le b_h\le C$; put
-$X(\Gamma,Ω)=average_{h\in I}b_h(\Gamma,Ω)$.
+$0le b_hle C$; put
+$X(Gamma,Ω)=average_{hin I}b_h(Gamma,Ω)$.
Then
-\int X\,d(\mu ⊗ \mathbb P_Ω)\asymp F\overline E,
+int X,d(mu ⊗ mathbb P_Ω)bbP_Ω)asymp Foverlineoverline E,E,
where the left side includes the level average and support
@@ -3513,11 +3513,11 @@ 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
+at $r$ alone suffices. Bounding the averaged cost $Xi$ of the two
band bridges uniformly conditional on the inner level and phases
gives
-\int X^2\,d(\mu ⊗ \mathbb P_Ω)\le CF\overline E\,\Xi.
+int X^2,d(mu ⊗ mathbb P_Ω)bbP_Ω)le CFoverlineoverline EE,Xi.
@@ -3535,22 +3535,22 @@ 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
+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
+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
@@ -3560,24 +3560,24 @@ attained outer endpoints and supports, integrate the outer completion
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}
+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
+$V_i+A_ir/R bmod 1$ is uniform and independent of $A_i$. Each
attained end therefore has probability at most
-$C\min(1,t/(1+g))$ 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
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, $\Xi\le Cl^{-1/2}$ on each window. Applying
+[
+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
Cauchy--Schwarz to (eq:sewing-first-moment) and
(eq:sewing-second-moment) yields
-$\overline E\le C\mathcal K\,\Xi/F$ and
-$B^*\ge cF\overline E/\Xi$, 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.
@@ -3586,34 +3586,34 @@ 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 $t=\min(t_1,t_2)$, $1\le t\le l$. 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 $r$,
+$C(1+min(g,t))^{-1/2}min{1,t/(1+g)}$. 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
+[
+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
gives
-\overline E\le C_\eta l^{3/4}(t/l)^\eta\mathcal K.
+overlineoverline EEle C_eta l^{3/4}(t/l)^etamathcal K.calK.
Indeed, the first moment on the target polygon measure is at least
-$cF\overline E$ and the second at most
-$CF\overline E\,l^{-1/2}(t/l)^\eta$, while $F\asymp l^{-5/4}$. For
+$cFoverlineoverline E$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 $C_\eta\sum_{j\ge0}(1+j)2^{-j\eta}<∞$; bins with
-$t/l\le\delta$ have total coefficient tending to zero as
-$\delta↓0$.
+$O(1+log(l/t))$ 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
+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
@@ -3623,13 +3623,13 @@ 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 $CR\ell_h$ per
+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
@@ -3639,38 +3639,38 @@ 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 $N\asymp R$ and
-\[
-\mathcal K\le CR^{-4/3}(R/d)^{15/8}.
-\]
+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
$P_1+P_2=d$,
-\sum_{eligible pairs}\mathcal K
-\le CR^2R^{-4/3}R^{15/8}\sum_{1\le d\le CR}d^{1-15/8}
-\le CR^{8/3}.
+sum_{eligible pairs}mathcal KcalK
+le CR^2R^{-4/3}R^{15/8}sum_{1le dle CR}d^{1-15/8}
+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
-\overline{\sum_{\gamma macroscopic}\ell(\gamma)^2w(\gamma)}
-\le CR^{41/12}.
+overline{sum_{gamma macroscopic}ell(gamma)^2w(gamma)}
+le CR^{41/12}.
-Bins with $l\le\delta R$ contribute at most
-$C\delta^{3/4}R^{41/12}+CR^{37/12}$; bins with $t\le\delta'R$ 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: $M_1\ge c_1R^{25/12}$. By Cauchy--Schwarz, a class with
+chords: $M_1ge c_1R^{25/12}$. By Cauchy--Schwarz, a class with
second moment at most $ε R^{41/12}$ contributes at most
-$C\sqrtε R^{25/12}$ to $M_1$. Choose $\delta,\delta'>0$ so
+$Csqrtε R^{25/12}$ 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.
@@ -3686,17 +3686,17 @@ $cR^{50/12}/R^{41/12}=cR^{3/4}$.
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
-$l\asymp R$, that estimate gives total bridge mass at least $cR^{3/4}$
+$lasymp R$, 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
-\[
-B_h\{\ell\ge aR^{4/3}\}\ge cR^{-1/4}
-\]
-for every admissible cap separation $hd_0\in[2AR,4AR]$. For
-arbitrary large $h$, choose $R\propto h$ so the selected caps lie in
+[
+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
these ranges; admissible rounding changes only fixed constants.
The strip first moment and completion of the proof.:
@@ -3724,10 +3724,10 @@ 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,
-\[
-\sum_{\gamma\in\mathcal B_h}(\ell(\gamma)-1)w(\gamma)\le Ch^{13/12}.
-\]
-The one missing centre per bridge adds only $B_h\le Ch^{-1/4}$,
+[
+sum_{gammainmathcal B_h}calB_h(ell(gamma)-1)w(gamma)le Ch^{13/12}.
+]
+The one missing centre per bridge adds only $B_hle Ch^{-1/4}$,
completing (eq:bridge-first-moment). The strip mass
(eq:bridge-mass) was proved in
Sections~c6:strip-transfer--c6:strip-asymptotics, and
@@ -3736,93 +3736,93 @@ Theorem~thm:finite-bridges is complete.
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