The honeycomb lattice H is the graph whose vertices are the centers of the
triangles of a unit triangular tessellation, with two vertices joined when the
triangles share a side. Three edges meet at every vertex. Euclidean distance,
diameter, and all geometric quantities refer to this embedding. Each triangle
of the tessellation has a center, and every walk below traverses triangle
centers. A port is the midpoint of a tessellation side on the boundary of a
region. A first-exit port-to-port walk in a region D is a self-avoiding walk
inside D with both endpoints on ports of D and no other vertex on a port of D.
A walk traversing N vertices has weight x^N where x = 1/sqrt(2+sqrt(2)) =
0.5411961... is the reciprocal of the honeycomb connective constant. Any
positive weight below is a product of such powers.
The connective constant of the honeycomb lattice equals sqrt(2+sqrt(2)),
established by
H. Duminil-Copin and S. Smirnov, The connective constant of the
honeycomb lattice equals sqrt(2+sqrt(2)), Annals of Mathematics 175
(2012), 1653--1665.
./pdf/signed_cylinder_pdfs/duminil-copin-smirnov-2012-1202.5523-honeycomb-connective.pdf
A polygon is a simple closed unoriented cycle on H; its length |lambda| is its
number of edges. Two polygons are incompatible when they share a vertex; each
is incompatible with itself. Polygon collections are summed with product
weights x^{sum |lambda|}.
Equilibrium has the following physical interpretation. Nienhuis's Coulomb-gas
analysis of the O(n) loop model predicted the two-dimensional self-avoiding
walk exponents, including nu = 3/4. The prediction is recorded in
B. Nienhuis, Exact critical point and critical exponents of O(n) models
in two dimensions, Physical Review Letters 49 (1982), 1062--1065,
and in the review
B. Nienhuis, Critical and multicritical O(n) models, Physica A 163
(1990), 152--157.
The exponent nu = 3/4 corresponds to reciprocal 1/nu = 4/3, which appears
below as the mean-first-exit-chord exponent and as the leading decay of the
marked two-bond sum. The empirical vanishing of critical bridges was proved
by Beaton, Bousquet-Melou, de Gier, Duminil-Copin and Guttmann in
N. R. Beaton, M. Bousquet-Melou, J. de Gier, H. Duminil-Copin and A. J.
Guttmann, The critical fugacity for surface adsorption of self-avoiding
walks on the honeycomb lattice is 1+sqrt(2), Communications in
Mathematical Physics 326 (2014), 727--754.
./pdf/signed_cylinder_pdfs/beaton-bousquet-melou-de-gier-duminil-copin-guttmann-2014-1406.3163.pdf
For honeycomb bridges Glazman and Manolescu obtained a logarithmic
subsequence bound, and for columnwise rhombic half-plane tilings they proved
invariance of boundary two-point functions; see
A. Glazman and I. Manolescu, Self-avoiding walk on Z^2 with Yang-Baxter
weights, Annales de l'Institut Henri Poincare B 56 (2020), 2281--2300.
Krachun and Panagiotis obtained a polynomial upper bound on the critical bridge
partition and used it to prove quantitative sub-ballisticity for uniform
honeycomb walks; see
D. Krachun and C. Panagiotis, Quantitative sub-ballisticity of
self-avoiding walk on the hexagonal lattice, Annals of Probability 54
(2026), 1109--1125.
The local mechanism on which the present argument depends is the parafermionic
cancellation of Duminil-Copin and Smirnov, extended to rhombic Yang-Baxter
weights by Glazman and Manolescu and by Ikhlef and Cardy:
Y. Ikhlef and J. Cardy, Discretely holomorphic parafermions and integrable
loop models, Journal of Physics A 42 (2009), 102001.
Three observables are treated. The first is a cylinder partition with two
marked points. Fix an even integer M = 2m, identify horizontal tessellation
sites modulo M, and mark the face centers O = (0,0) and O' = (m,0) on the cut.
The cut sites are split into consecutive lists A (from O to O') and B (from
O' back to O), each of size m. The cylinder is infinite vertically and is
defined by the increasing height limit of finite cylinders with both rims
closed (every port on either rim unused). A separating cycle is a simple
closed cycle of H whose complement places O and O' in different components.
Let G_m(g) be the sum over families of pairwise disjoint separating cycles of
prod_{lambda in family} g x^{|lambda|},
the empty family contributing 1. Because a separating cycle crosses the cut
between the marked points, G_m(g) has degree at most 2m in g.
The second observable is the chord law inside a regular tessellation hexagon
D_R of side R. Let Z(R) be the sum over collections of disjoint polygons
inside D_R that surround the central face, with weight 2 x^{|lambda|} per
polygon and empty collection contributing 1. Let C_R be the set of first-exit
chords of D_R of diameter at least R/10, where a chord is a first-exit walk
whose endpoints are boundary ports of D_R, the sum runs over every starting
port and every possible first-exit port, and the direction from start to exit
is retained. Define
Z_R^chord = sum_{gamma in C_R} x^{N(gamma)},
P_R^chord(gamma) = x^{N(gamma)} / Z_R^chord.
The chord length N(gamma) is the number of vertices of gamma.
The third observable is the two-bond sum for planar polygons. For H > 0, put
S(H) = sum_{diam lambda <= H} x^{|lambda|} |lambda|^2,
summed over polygons modulo lattice translations. The squared length factor
|lambda|^2 counts ordered pairs of bonds inside lambda; the sum over such
pairs decomposes S(H) into a sum over two-bond configurations, whose control
is the content of the third theorem.
Through all statements, F(R) = R^{a+o(1)} means that log F(R) / log R
converges to a, an upper bound F(R) <= R^{a+o(1)} means: for every eps > 0
there are C_eps and R_eps such that F(R) <= C_eps R^{a+eps} for all
R >= R_eps, and a lower bound is analogous. A bound is uniform in some data
when the constants C_eps and thresholds R_eps do not depend on that data.
Theorem (cylinder pressure and planar nesting). As m -> infinity through even
positive integers, the balanced cylinder partition with two marks satisfies
G_m(2) = m^{1/6+o(1)},
-(d^2/dg^2) log G_m(g) at g = 0 is O(log m),
the derivative being the total weight of ordered pairs of incompatible
separating cycles (pairs sharing at least one vertex, including each cycle
with itself), and locally uniformly for complex Theta with |Re Theta| < pi,
|G_m(2 cos Theta)| <= m^{Re(1/6 - 2 Theta^2/(3 pi^2)) + o(1)}.
The planar nesting partition satisfies Z(R) = R^{1/12+o(1)} as R -> infinity
through integers.
Theorem (finite chords and marked polygons). For the chord law of D_R,
Z_R^chord = R^{3/4+o(1)},
E_R^chord N = R^{4/3+o(1)},
and uniformly for honeycomb vertices v at distance at most R/10 from the
central face,
P_R^chord(v lies on gamma) = R^{-2/3+o(1)}.
Moreover,
S(H) <= H^{2/3+o(1)}.
Theorem (unequal marked intervals). On a physical tilted cylinder whose
unmarked lists have m+1 and n+1 sites, with n >= m and M = m+n+2 even, with
unmarked logarithmic heights -f and 1-f where 0 <= f <= 0.6, and with site
counts differing from m(1-f), mf and n(1-f), nf by at most a fixed constant,
let H^poly be the total critical weight of a single unoriented polygon
crossing two horizontal-type marked bonds. Then uniformly over the lists and
orders, for every eps > 0 there are C_eps and m_eps such that
H^poly <= C_eps n^{eps} m^{-4/3} (n/m)^{1/3}
for all n >= m >= m_eps.
The remainder of the paper proves these statements in an interleaving that
mirrors their mechanism. The boundary balance and local transfer identities
are established first, since they seed the local cancellation and the finitude
of vacuum limits. The half-plane arch estimate then controls boundary mass.
The cylinder theorems proceed through an oriented transfer, Yang-Baxter and
fusion identities, a finite spin sum, contour and bosonization rewritings, an
absolute stability estimate on the resulting signed gas, and a compact
propagator that isolates the physical power. The chord and visit corollaries
follow by combining the arch estimate, a boundary visit identity, and the
planar nesting exponent. The two-bond estimate is proved independently, with
an analogous signed-transfer pipeline but with two marked bonds instead of one
unmarked cycle.
Throughout, s = 3/8 and alpha = pi/8, so 2 x cos(s pi / 3) = 1. The seed is
the parafermionic turn balance of Duminil-Copin and Smirnov.
Lemma (convex balance). Let D be any finite convex tessellation polygon, and
let a be a port on the boundary of D. The signed sum over first-exit port-to-
port walks gamma of D from a, weighted by x^{N(gamma)} e^{i s T(gamma)}, where
T(gamma) is the total turning of gamma in radians and orientation is counted
with sign, equals one.
Proof. Start with the nonbacktracking tree of walk continuations from a,
with an outgoing factor x e^{i s pi/3} or x e^{-i s pi/3} at each vertex
depending on whether the walk turns left or right. The two outgoing factors
sum to 2 x cos(s pi / 3) = 1 at every interior vertex. The tree stops the
first time an already-visited vertex is about to be entered. Every leaf of
this tree is one of the following: a boundary port (that is, a first-exit
walk, which is the quantity being summed), or a first self-collision loop,
closed by the incoming stem.
A first self-collision closes a simple loop. In the flat annular region
formed by the loop together with the stem, the stem approaches the loop from
outside. The loop's turn contribution is either +5pi/3 or -5pi/3 (the loop
itself has turn +-2pi, but one turn of pi/3 at the closure corner has been
dropped because only two of the three branches at that interior vertex are
used), and the stem's turn onto the loop is -+pi/3. The two orientations of
the loop are summed with weight e^{i s(+-5pi/3 -+ pi/3)}. The total turn per
orientation is +-4pi/3, so the combined phase contribution at the self-
collision is 2 cos(4 pi s / 3) = 2 cos(pi / 2) = 0. Every self-collision
therefore cancels. Only first-exit walks remain, and the total signed sum
equals the same factor that the interior vertices put at each turn, which is
one.
For a first-exit walk to a port b, the total turning T(gamma) equals Delta -
pi, where Delta is the counterclockwise boundary tangent turn accumulated
from a to b along the boundary, together with the two extra quarter turns
needed to close the polygon. Each boundary traversal below contributes
cos(s T) >= sin(alpha) > 0 because Delta is a boundary turn plus a fixed
compensation. Taking real parts of the balance and using the positivity gives
the finiteness of the total positive weight of first-exit walks. QED
Let H(d) be the total positive weight x^N of first-exit walks on the half-
plane above a horizontal tessellation line, from a fixed boundary port to the
port d steps to the right. Reflection in the vertical axis gives the same
total to the left. Taking real parts of the convex balance in the half-plane
and using the two equal sides, gives
2 sin(alpha) sum_{d>0} H(d) <= 1,
with strict equality in the limit and no contribution from d = 0 (a first-exit
arch has positive length). In particular sum_{d>0} H(d) <= 1/(2 sin alpha) <
3/2, so the sum converges.
The following transfer-matrix framework realizes the balance identically as a
matrix-element equation. Consider a cylinder obtained by identifying modulo M
steps horizontally, with rows tiled by M rhombi of basis 1 and e^{i theta},
with theta potentially varying by row. Inside each rhombus, connections
(possibly empty) pair up bond midpoints on distinct sides, using each midpoint
at most once, noncrossingly. The empty weight is 1, a straight connection
between opposite sides weighs v, single turns of magnitudes theta and
pi - theta weigh u and d respectively (where port directions are normal to
sides), and two simultaneous turns weigh a for the pairing left-down and
up-right, or b for left-up and right-down. Write phi = 3 theta / 8 and let
s_k = sin(phi + k alpha), h = sin(2 alpha). The trigonometric SAW weights are
(u, d, v, a, b) = (h s_5, h s_0, s_0 s_5, s_5 s_6, -s_0 s_7) / (s_2 s_3).
At phi = alpha these are (x, x^2, x^2, x^2, 0), exactly the honeycomb weights
after splitting each rhombus into the two equilateral triangles.
For a finite stack, close both rims. The partition Z of one unoriented
noncontractible polygon is the linear coefficient in a fugacity n of the full
loop sum assigning weight zero to contractible loops and weight n per
noncontractible loop. For real angles 0 < theta < pi, orient each loop with a
factor exp(-i T / 4) and twist p to the signed number of seam crossings (east
positive); summing over the two orientations produces the loop weights n =
p + p^{-1}. This is because an embedded loop in the flat cylinder is either
contractible with turn +-2 pi, or has winding +-1 and turn zero. The turning
identities then extend to the rhombic model algebraically.
The transfer matrix. On each horizontal-flow and descending-flow bond put a
spin in {0, +1, -1}. The oriented matrix acts with row indices (left, up) and
column indices (right, down), with positive signs meaning left-to-right or
up-to-down on the corresponding bond. After removing the telescoping phase
e^{i theta (left - right) / 4}, the resulting R(phi) is block-diagonal with
respect to the total sum s_left + s_up = s_right + s_down = J. Writing
lambda = e^{-2 i alpha}, the blocks are
(s', s'), s' = +-1 : [a]
(s', 0), (0, s') : [[v, u], [u, v]]
(1, -1), (0, 0), (-1, 1) : a 3 x 3 block
[[b, d lambda, b lambda^2 + a],
[d lambda^{-1}, 1, d lambda],
[b lambda^{-2} + a, d lambda^{-1}, b]].
The turns connecting left to down have magnitude theta (weights u, d, a, b as
above); those connecting left to up have magnitude pi - theta (weights v, u,
d, b, a with the substitutions s_j -> s_{j+4} of the trig functions).
Exchange. Let V = C^3 with basis |0>, |+>, |->, and let subscripts index
ordered tensor factors. On a copy V_i let s_i act by s_i |k> = k |k>. Write
P_{ij} for interchange of factors i, j, and put Delta_R(psi) = sin(psi + 2
alpha) sin(psi + 3 alpha).
Lemma (local exchange and fusion). The entries of R(psi) are meromorphic,
with possible poles at Delta_R(psi) = 0, and the following are identities of
meromorphic matrix functions:
R_{12}(t) R_{13}(t + phi) R_{23}(phi) = R_{23}(phi) R_{13}(t + phi) R_{12}(t),
R_{12}(phi) R_{21}(-phi) = I.
In particular R_{12}(0) = P_{12}.
Proof. The unitarity R_{12}(phi) R_{21}(-phi) = I follows by block multipli-
cation, using lambda^4 = -1 and the elementary relation aa' = 1, ab' + ba' +
dd' = 0, ad' + d = a'd + d' = 0, vv' + uu' = 1, vu' + uv' = 0, where primes
denote the same quantity at -phi. Each displayed scalar identity reduces to
sin_j' = sin_{8-j} and product-to-sum.
To prove the Yang-Baxter exchange at the three special values t = 0, 2 alpha,
3 alpha, it suffices to check the factorizations R_{12}(0) = P_{12},
R_{12}(2 alpha) = J_2 J_2^*, R_{12}(3 alpha) = J_3 J_3^*, together with the
three-space identity in each block; the block calculations are routine. For
general t the exchange identity is proved by noting that, upon multiplying
the exchange difference by the product of the three Delta_R factors appearing
(t, t + phi, phi), every entry becomes a polynomial in e^{2 i t} of degree at
most 4. The six values 0, 2 alpha, 3 alpha, -phi, 2 alpha - phi, 3 alpha - phi
are distinct modulo pi for generic real phi, so this polynomial of degree at
most 4 vanishes if it vanishes at these six values. Each of the six reduces
by the swap symmetry R(psi + pi) = Z_1 R(psi) Z_1 where Z |k> = (-1)^{k^2}
|k>, to one of the factorizations at 0, 2 alpha, 3 alpha. Continuation in the
two parameters proves the meromorphic identity. QED
The exchange identity has a consequence about the family of rows: rows of any
quantity whose entries are R-matrices with a common twist have a simultaneous
eigenbasis, and their eigenvalues depend only on the row's rapidity. Fusion
extends this to admitting a row to be collapsed.
Fusion. Two adjacent rows can collapse when their rapidities differ by a
quantized amount. Set E_2 = V and E_3 = C, and define injective maps
J_delta : E_delta -> V tensor V by
delta = 2:
J_2 |0> = |00> + x sum_{k = +-1} lambda^k |k, -k>
J_2 |k> = x (|k, 0> + |0, k>) for k = +-1
delta = 3:
J_3 (1) = sum_{k = 0, +-1} lambda^k |k, -k>.
On E_2 put s_f |k> = k|k>, and on E_3 put s_f = 0.
Lemma (fusion factorization). For delta = 2, 3 the identity
R_{12}(delta alpha) = J_delta J_delta^*
holds in V tensor V, and for any extra factor V_k the meromorphic site
intertwining holds:
R_{23}(phi) R_{13}(phi + delta alpha) (J_delta tensor I)
= (J_delta tensor I) R_f(phi)
R_{a1}(psi) R_{a2}(psi + delta alpha) (I_a tensor J_delta)
= (I_a tensor J_delta) R_f(psi + [alpha for delta = 2 else 0])
with R_f(phi) = R(phi + alpha) for delta = 2 and R_f = I for delta = 3.
Proof. At the three values psi = 0, 2 alpha, 3 alpha the five weights are,
respectively,
psi = 0 : (1, 0, 0, 1, 0)
psi = 2 alpha : (x^2, x, x^2, 0, x^2)
psi = 3 alpha : (0, 1, 0, 0, 1).
The R(psi) block decompositions at these values produce the depicted images
directly. For instance, the charge-zero column of J_2 is (x lambda, 1, x
lambda^{-1})^T, and the charge +-1 columns are x (1, 1)^T; substituting into
J_2 J_2^* reproduces R_{12}(2 alpha). The factorizations are not isometries:
J_2^* J_2 = diag(3 - sqrt 2, 2 - sqrt 2, 2 - sqrt 2) in the standard basis,
and J_3^* J_3 = 3.
The three-space intertwining is verified by explicit block multiplication in
each charge sector. The scalar identities needed reduce, after cancelling
common factors, to a finite list of sine product-to-sum identities together
with lambda^2 + lambda^{-2} = 0; the block-by-block check is direct. The
row-space formula j:site-fusion is the transpose-in-spaces image of the
two-space version, obtained by conjugating the row intertwining and exchanging
factors 1 and 2, using the invariance of J_delta under complex conjugation in
real rapidities, and then continuing meromorphically. QED
Lemma (restriction to fusion image, and quotient at fusion). Fix delta in
{2, 3} and put W_delta = im J_delta subset V_1 tensor V_2. For an extra factor
V_k set F_{delta, k}(psi) = R_{2k}(psi) R_{1k}(psi + delta alpha). At regular
parameters the subspace W_delta tensor V_k is invariant; transported through
J_delta tensor I the restriction reproduces R_f(psi). At psi = 0 the full
image of F_{delta, k}(0) lies in W_delta tensor V_k, so the induced operator
on ((V_1 tensor V_2) / W_delta) tensor V_k is zero at psi = 0 and holomorphic
near psi = 0 (being a polynomial in the local parameter divided by Delta_R
factors that do not vanish there). The twist restricts by
p^{s_1 + s_2} J_2 = J_2 p^{s_f} and p^{s_1 + s_2} J_3 = J_3 for every p != 0.
Proof. The invariance and the action on the image are the site intertwining
of the previous lemma, transported through the identification E_delta -> W_delta
supplied by J_delta. At the fusion point the factorization supplies the
stronger inclusion
F_{delta, k}(0) = P_{2k} R_{1k}(delta alpha)
= P_{2k} J_{delta, 1k} J_{delta, 1k}^*,
im F_{delta, k}(0) subset W_delta tensor V_k,
where J_{delta, 1k} is the embedding applied on factors 1 and k. Indeed
P_{2k} carries im J_{delta, 1k} onto im J_{delta, 12} = W_delta. The quotient
operator is holomorphic at psi = 0 because the local parameters 0, delta alpha
are regular; its vanishing at psi = 0 then factors as (z - 1) times a
holomorphic function in the local coordinate z = e^{2 i psi}. The twist
identities follow because every term of J_2 |s_f> has total spin s_f, and
every term of J_3 has total spin zero. QED
Fourier kernel. The finite-vacuum normalization and the boundary moment
problem both use the same elementary transform.
Lemma (Fourier kernel). For 0 < beta < pi and |Im y| < pi - beta,
int_R e^{i k y} (sinh(beta k) / sinh(pi k)) dk
= sin(beta) / (cosh y + cos beta).
The integral and all parameter derivatives converge locally uniformly on
the indicated strip.
Proof. For real y > 0 close the contour by rectangles in the upper half-plane
with upper edges at half-integer heights. The ratio of hyperbolic sines decays
exponentially on the vertical edges, and the factor e^{i k y} controls the
upper edge as its height tends to infinity. The residues at k = i n (n >= 1)
give
2 sum_{n >= 1} (-1)^{n + 1} sin(n beta) e^{-n y}
= sin(beta) / (cosh y + cos beta).
The geometric-series evaluation proves the last equality. Evenness extends it
to negative y, and dominated convergence covers y = 0. Exponential decay in
|k| proves local uniform convergence and parameter differentiation on smaller
strips. Analytic continuation in y gives the asserted domain. QED
The transfer formalism, the convex balance, and the fusion identities seed
everything that follows. Their combination produces a signed-transfer
representation of both the cylinder partition and the two-bond sum. The
remaining work is to make the representation's individual signed terms
summable and to read off the exponents from the absolute sum.
The half-plane arch estimate. Fix a port on the boundary of a horizontal
tessellation half-plane. Let W_R be the total critical weight x^N of first-
exit same-boundary arches reaching Euclidean distance at least R from that
port, and for d > 0 let H(d) be the total weight of first-exit arches from
the fixed port to the port d boundary steps to its right.
Proposition (boundary arch mass). W_R = R^{-1/4+o(1)}, and
2 sin(pi/8) sum_{d>0} H(d) = 1,
sum_{d >= n} H(d) <= n^{-1/4+o(1)}.
Proof.
The fused cylinder increment. On a finite cylinder of circumference M with
closed rims, let Z be the partition of exactly one unoriented noncontractible
polygon. Write T_p(phi) for the oriented row transfer matrix with row
parameter phi, and put alpha = pi/8. Row matrices with the same twist commute:
multiplying the local exchange identity across the sites and tracing over the
two auxiliary spaces, the twist p^{s_1 + s_2} commutes with the exchange
matrix. Then they commute wherever that matrix is invertible, and identically
by continuation.
Set U = im J subset V_1 tensor V_2 for the embedding at 2 alpha, and at site
j put X_j(phi) = R_{2j}(phi) R_{1j}(phi + 2 alpha). Locally this preserves
U tensor V_j and induces R(phi + alpha) through the isomorphism J; at phi = 0,
X_j(0) = P_{2j} R_{1j}(2 alpha), im X_j(0) subset U tensor V_j,
because R_{1j}(2 alpha) = J_{1j} J_{1j}^* and the swap carries embedded
spaces 1, j to 1, 2. The operator induced on the quotient by U tensor V_j is
O(phi), so its product over M sites is O(phi^M). Taking the auxiliary trace
gives
T_p(phi) T_p(phi + 2 alpha) = T_p(phi + alpha) + O(phi^M), (*)
locally holomorphic in (phi, p) near (0, i). Extract the coefficient of n =
p + p^{-1}: at p = i, dn/dp = 2, which preserves the order of vanishing.
Now add one new outer row on L honeycomb rows with all parameters alpha and
subtract the partition without the new row. Let H_M(d) be the weight of half-
plane arches from a fixed port to the port d steps right, disjoint from all
their nontrivial horizontal translates by M. For fixed M these converge as
L -> infinity, since dropping mutual disjointness bounds each by products of
finitely many H(d) and the coarse summability from the convex balance applies.
The tile weights satisfy u d = c_alpha v (1 - v) with c_alpha = 2 sin alpha.
A k-interval term has at most M - 2k straight tiles, so the limiting increment
is a polynomial F(v) of degree at most M, and
F(v) = v^M + M c_alpha v (1 - v) sum_{d = 1}^{M-1} v^{M - d - 1} H_M(d)
+ (1 - v)^2 Q_M(v),
F(1) = 1,
F'_v(1) = M (1 - c_alpha sum_{d = 1}^{M-1} H_M(d)),
for a polynomial Q_M. Inclusion--exclusion for two added rows, and domination
by positive honeycomb weights, pass (*) to the limit. With y = i(2 phi - 3
alpha) and v = (cosh y - sin alpha) / (cosh y + cos alpha), the resulting
identity near y = -3 i alpha is
f(y) + f(y + 4 i alpha) - f(y + 2 i alpha) = 1 + O((y + 3 i alpha)^M),
where f(y) = 1 - F(v(y)).
Positive moment representation. Index columns j = 0, ..., M - 1, and set
A_j(k) = partial_alpha^j sinh(alpha k),
B_j(k) = partial_alpha^j cosh(alpha k),
w(k) = 2(1 + 4 sinh^2(alpha k)) / sinh(pi k),
w_0(k) = k w(k).
Since F(1) = 1, the function f(y) is a linear combination of (cosh y + cos
alpha)^{-r}, 1 <= r <= M. The first M parameter derivatives of sin alpha /
(cosh y + cos alpha) = int_R e^{i k y} sinh(alpha k) / sinh(pi k) dk span
these powers: the derivative of order j has a nonzero coefficient of the
denominator power j + 1. The Fourier kernel lemma gives a vector c with
f(y) = int_R e^{i k y} A(k)^T c / sinh(pi k) dk,
and the integral and required derivatives converge locally uniformly in
|Im y| < pi - alpha. At y = -3 i alpha, splitting positive and negative k gives
the moment matrix equation
D c = B(0), D = int_0^infty B(k) A(k)^T w(k) dk.
D is nonsingular. The ordered determinant expansion of det D is an integral
over 0 < k_1 < ... < k_M of a product of two determinants times positive
weights. Both factors are strictly positive: expanding sinh or cosh in
positive power series gives each via a positive generalized Vandermonde, and
the sign of the second follows by separating the positive k_i in scale. Then
det D = int over configurations of (positive factor) times Delta(k)^2, hence
> 0.
Boundary residue. Set K(z) = A(z)^T D^{-1} B(0). For real y -> +infinity,
the residue at k = i of the Fourier formula for f gives -2 i K(i) e^{-y} as
its first term. On the polynomial side,
1 - v = 2(cos alpha + sin alpha) e^{-y} + O(e^{-2 y}),
so
F'_v(1) = -i K(i) / (cos alpha + sin alpha).
Orbital form. The positive density
det[A(k_1), ..., A(k_M)] det[B(k_1), ..., B(k_M)] prod_i w(k_i)
on 0 < k_1 < ... < k_M is realized by the Harish-Chandra--Itzykson--Zuber
integral: for D_k = diag(k_1, ..., k_M, -k_1, ..., -k_M) and Q = U diag(alpha
I, -alpha I) U^* with U Haar on U(2M), put I(k) = E exp(tr(D_k Q)). Then the
density is proportional to
Delta(k^2)^2 prod_i w_0(k_i) I(k),
which is a positive probability density. The derivation from the HCIZ formula
follows by compressing U D_u U^* to a (2M - 1)-dimensional minor; the ordered
eigenvalues interlace, the last resolvent residue gives uniform simplex
weights, and the resulting Jacobian is Delta(l) / Delta(u). Induction with
test parameters z_i - z_d and column subtraction produce the paired
determinantal formula. Specializing to paired lists (k_1, ..., k_M, -k_1,
..., -k_M) and (u_1, ..., u_M, -u_1, ..., -u_M), using the paired Vandermonde
|Delta(+-k)| = 2^M (prod_i k_i) Delta(k^2)^2,
and splitting the exponential determinant into sinh and cosh blocks gives the
form. The orbital density extends continuously when one point approaches
zero, and it is integrable:
w_0(k) ~ 4 k e^{-6 alpha k} (k -> infinity), w_0(0) = 2 / pi.
Pinning. Let rho(0) be the one-point intensity at zero of the orbital law.
Pinning one point there leaves N = M - 1 positive points with density pro-
portional to Delta(k^2)^2 prod_{j = 1}^N k_j^4 w_0(k_j) I(k_1, ..., k_N, 0),
where the orbital integral still uses the 2M-dimensional orbit with one zero
pair in D_k. For its expectation E_pin,
sin alpha / alpha * rho(0)/w_0(0) * E_pin prod_j (1 + k_j^{-2})
<= -i K(i)
<= rho(0)/w_0(0) * E_pin prod_j (1 + k_j^{-2}).
The pinning proof compares determinants with one evaluated at the missing
zero, using a Bessel function identity
(sinh(k_j t)/f_z(t))' = ((k_j^2 - z^2) / f_z(t)^2)
int_0^t f_z(v) sinh(k_j v) dv
for f_z(u) = sinh(z u)/z, so that ratios of determinants reduce to positive
weighted averages. The averaging weights satisfy the elementary bounds
g(u_M) <= R <= g(u_1)/g(u_M) for g(t) = sin t / t, giving the displayed
two-sided pinning bound.
Orbital derivatives and stochastic comparison. For either orbital factor,
there are q_j >= 0 with partial_{k_j} log I = 2 q_j and
0 <= q_j <= min{alpha, alpha^2 k_j, alpha^2 / sum_{ell} (k_j + k_ell)^{-1}}.
Derivation: set q_j = under the tilting exp(tr(D_k Q)), exchange
all paired positions and send Q to -Q to get = -q_j. Convexity in
k_j from the log-moment-generating structure gives q_j >= 0, and ||Q|| = alpha
gives q_j <= alpha. Infinitesimal conjugation gives
= (D_k{aa} - D_k{bb}) <|Q_{ab}|^2>; for a = +j, b = -ell
this becomes <|Q_{+j, -ell}|^2> = (q_j + q_ell)/(k_j + k_ell). Summing over
all ell and using sum_b <|Q_{+j, b}|^2> = alpha^2 (from Q^2 = alpha^2 I) and
dropping q_ell >= 0 gives the third bound.
The orbital density's log-derivative relative to the pure density with weight
k^e w_0(k) e^{2 alpha k} is 2 q_i - 2 alpha <= 0; positive association of the
underlying pure law (Vandermonde determinant squared) then gives
O_N^{(e)} <=_st P_N[k^e w_0(k) e^{2 alpha k}], e = 0, 4,
and the first j such points are stochastically below P_j[k^e w_0(k) e^{2 alpha
k}]. The logarithm of w_0 has derivative D_0(k) satisfying
-6 alpha < D_0(k) < 0,
D_0(k) -> -6 alpha (k -> infinity),
D_0(k) = -(40/3) alpha^2 k + O(k^3) (k -> 0).
Consequently, with m = floor(N^{1 - epsilon}), on an event E of probability
1 - O(e^{-c m}) the small-index coordinates satisfy
sup_{i <= m, k in E} q_i <= C / (epsilon log N) = o(1).
Gamma comparison and evaluation. For ell, beta > 0 let W_{ell, beta}(k) =
|Gamma(ell + i beta k)|^2 and b = pi beta. For every fixed b_+ > 6 alpha there
is ell_+ > 0 such that with beta_+ = b_+/pi,
P_N[k^e W_{ell_+, beta_+}] <=_st O_N^{(e)} for all N, e = 0, 4.
For fixed 0 < b_- < 6 alpha and 0 < epsilon < 1 there is ell_- > 0 such that,
with beta_- = b_-/pi and for sufficiently large N,
L_{O_N^{(e)}}(k_{<= m} | E, k_{> m}) <=_st P_m[k^e W_{ell_-, beta_-}]
almost everywhere on later configurations compatible with E. The Gamma
derivative
G_{ell, beta}(k) = -2 beta^2 k sum_{n >= 0} 1/((n + ell)^2 + beta^2 k^2)
satisfies G < -b + 2/k for 0 < ell <= 1 and G >= -b with |G|/k bounded by
2 beta^2 (ell^{-2} + ell^{-1}) for ell >= 1. Comparing G with D_0 gives the
two stochastic inequalities on bounded intervals and tails.
Evaluate the pure statistic on the Meixner--Pollaczek law. Normalize
|Gamma(ell + i u)|^2 du to a probability on R; its Fourier transform is
cosh(t/2)^{-2 ell}. The generating function (1 - i z)^{-ell + i u} (1 + i z)^{-
ell - i u} = sum_n P_n(u) z^n gives real polynomials P_n of degree n with
leading coefficient 2^n/n!, parity n, and squared norms c_n = (2 ell)_n/n!
~ n^{2 ell - 1}/Gamma(2 ell). Evaluations:
(-1)^j P_{2j}(0) ~ j^{ell - 1}/Gamma(ell),
(-1)^j P_{2j}(i beta) ~ 2^{2 beta - 1} j^{ell + beta - 1}/
Gamma(ell + beta).
The one-point intensity at zero and the pinned statistic give
rho_{Gamma, L}(0) ~ 1 + log L,
E_{Gamma, L}^pin prod_j (1 + k_j^{-2}) ~ L^beta / (1 + log L).
For the unpinned orbital law, the two-sided pin bound and the Gamma
comparisons give
c <= rho(0) <= C (1 + log M),
with constants depending on the comparison parameters. For the pinned law,
the product is decreasing, so the first Gamma comparison gives a lower bound
of the pure statistic; conditioning on E and discarding later factors at
least one gives the opposite, and P(E) = 1 - O(e^{-c m}) closes the loop.
Order of limits. Fix b_+ > 6 alpha for the upper, b_- < 6 alpha and epsilon >
0 for the lower, choose their ell parameters, and only then let M -> infinity.
Put a_M = log F'_v(1) / log M. The pinning and residue bounds give
(1 - epsilon) b_-/pi <= liminf a_M <= limsup a_M <= b_+/pi.
Letting b_-, b_+ -> 6 alpha and epsilon -> 0 gives
F'_v(1) = M^{3/4 + o(1)}. (**)
Deficit and lower exponent. Define the positive deficit
delta_M = 1/c_alpha - sum_{d = 1}^{M-1} H_M(d) = F'_v(1) / (c_alpha M)
= M^{-1/4 + o(1)}.
The coarse bound c_alpha sum_{d > 0} H(d) <= 1 from the convex balance, the
inequality H_M(d) <= H(d), and delta_M -> 0 give saturation sum_{d > 0} H(d) =
1/c_alpha. With A(n) = sum_{d >= n} H(d), this yields A(M) <= delta_M. Every
arch contained within distance strictly less than M/2 of its source is
disjoint from its nontrivial translates by M, so arches counted in the deficit
reach distance at least M/2; hence delta_M <= W_{M/2}. Choosing M just above
2 R gives W_R >= R^{-1/4+o(1)} and A(n) <= n^{-1/4+o(1)}.
Wedge comparison. Intersect the upper half-plane with the wedge between rays
of angles 0 and theta = pi/3. A source on its horizontal ray has horizontal
distance a in {1/2, 3/2, ...} from the vertex. The signed balance holds on
the infinite wedge: exhaust the half-plane by convex tessellation polygons
with extending horizontal bases, use the convex-balance positivity for same-
side arches, and pass to the limit using the absolute convergence of the fixed-
ray exit sums.
Put s = 3/8. For same-side walks the phase is e^{i s pi} to the left and
e^{-i s pi} to the right, and exits to the other ray have phase e^{i s theta}.
Let L_<(a) and L_>(a) be the lost left-side and right-side same-side arch
weights on restriction from the half-plane to the wedge, including endpoints
removed by the restriction. Subtracting the balances and projecting onto
e^{i s theta} gives
sin(s(pi - theta)) L_<(a) - sin(s(pi + theta)) L_>(a) = 0,
so L_<(a) = r L_>(a) with r = sqrt 2. For positive half-integer ports a, b,
let D(a, b) be the lost same-side weight with those fixed endpoints. Reversal
gives D(a, b) = D(b, a). The missing negative ports to the left contribute
A(ceiling a), so
L_<(a) = A(ceiling a) + sum_{0 < b < a} D(a, b),
L_>(a) = sum_{b > a} D(a, b).
Summing L_< = r L_> over 0 < a <= T gives
(r - 1) sum_{0 < a < b <= T} D(a, b) + r sum_{0 < a <= T < b} D(a, b)
= sum_{0 < a <= T} A(ceiling a)
<= T^{3/4 + o(1)}.
Since r > 1, this bounds sum_{0 < a <= T} (L_<(a) + L_>(a)) by the same order.
After translating the source to a fixed port, the wedges increase with a, so
their total loss decreases and
L_<(a) + L_>(a) <= a^{-1/4 + o(1)}.
Two reflected wedges whose vertices are horizontal distance a on opposite
sides of the source intersect in a triangle of diameter O(a). An arch reaching
distance C a for a sufficiently large fixed C must leave at least one wedge,
so its weight is at most the sum of the two losses, giving W_R <=
R^{-1/4+o(1)}. This completes the boundary arch mass proposition. QED
Extremal segments. Lemma. Fix a honeycomb vertex and two distinct height
levels, with the fixed vertex on one level. The total critical weight of SAW
segments from that vertex to any vertex on the other level, whose endpoints
are opposite height extrema of the segment, is bounded by an absolute constant
independent of the levels.
Proof. Extend each endpoint to a port on the adjacent horizontal line beyond
its extremal level. This requires at most one additional triangle at each
end, strictly beyond the segment's range of heights: between two successive
horizontal lines the lower triangles have a port on the bottom and the higher
triangles have a port on the top; a triangle of either type has a neighbor of
the other type. Choose an extension by a fixed local rule. Trimming recovers
the segment, and possible choices have bounded multiplicity. The extended walk
is a first-exit crossing of the slab between the two chosen lines. The convex
boundary balance, exhausted horizontally, bounds its total weight for each of
the finitely many possible starting ports. Removing the at most two extra
vertices costs at most x^{-2}. QED
For later use, split an n-edge walk at a highest vertex and treat the two
halves. Within either half, go to the last occurrence of its lowest remaining
level, then to the last occurrence of its highest remaining level, and continue
alternately. These are segments of the lemma. Their positive integer spans
decrease strictly after a possible equality of the first two, since each
chosen extremum is the last occurrence of its level. The spans sum to at most
2 n, so there are O(sqrt n) segments and at most (2 n + 1)^{O(sqrt n)} span
lists. For a prescribed list, successive use of the lemma bounds the endpoint-
summed product of segment weights by a constant to the number of segments,
and recovering the weight of the joined walk costs x^{-1} per shared endpoint.
These subexponential concatenation bounds will be used below in the marked
argument, and they also recover the reciprocal exponential growth rate of
full-plane SAW by summing over highest-vertex splits.
Chords on the boundary. Let K be a finite convex tessellation polygon D
sharing a starting port with the same inward direction as a containing convex
cap D_0. Let L be the raw weight of first-exit chords of D_0 other than those
confined to K and ending at a boundary port shared by K and D_0, and let E be
the raw weight of first-exit chords of K ending on ports newly created by the
restriction. Then
sin(alpha) L <= E <= L / sin(alpha).
Proof. The common chords, confined to K and ending on a shared true boundary
port, have the same intrinsic turning phase in both balances and cancel. The
two remaining complex sums are equal. Every real coefficient in either convex
domain lies between sin(alpha) and 1. Taking real parts proves both
inequalities. The case with D_0 the saturated half-plane and K a finite convex
cap follows by the same convex exhaustion used in the wedge argument. QED
Visit identity. Let D be a finite convex tessellation polygon and v a
honeycomb vertex not adjacent to a boundary port. Sum over all ordered first-
exit boundary chords of D that visit v and set
S_D(v) = sum_{gamma ni v} x^{N(gamma)} cos(3 T(gamma)/8).
Let Z_v be the partition of disjoint unoriented cycles in D strictly enclosing
v and avoiding v, with weight 2 x^{|lambda|} per cycle. Let Z_out(lambda) count
disjoint cycles in D surrounding and disjoint from lambda with the same
weights. At a vertex of lambda at v, call the corner convex or reflex when its
interior angle is 2 pi/3 or 4 pi/3, and put
c_lambda = sqrt 2 when v is a convex corner of lambda,
= 2 + sqrt 2 when v is a reflex corner.
Proposition. The identity
S_D(v) = 3 Z_v + sum_{lambda ni v} c_lambda x^{|lambda|} Z_out(lambda)
holds. Moreover, letting Z_D(f) denote the analogous partition around a face
center f for each of the three incident face centers,
sum_{f ~ v} Z_D(f) = 3 Z_v + sum_{lambda ni v} 2 k_lambda x^{|lambda|}
Z_out(lambda),
with k_lambda = 1 at a convex corner and 2 at a reflex corner, and therefore
(1/sqrt 2) sum_{f ~ v} Z_D(f) <= S_D(v) <= sum_{f ~ v} Z_D(f),
S_D(v) <= sum_{gamma ni v} x^{N(gamma)} <= S_D(v) / sin(alpha).
Proof. Remove the triangle centered at v, creating three inner ports. For
each outer starting port the signed balance still holds in the punctured
domain: its source is outside every possible first-collision loop, including
a loop surrounding the removed triangle, so the same collision cancellation
applies. Subtracting the punctured balance from the original one cancels
outer chords avoiding v, leaving the outer chords through v equal to the paths
ending at an inner port. Take real parts, sum over the outer sources, and
reverse the paths; reversal changes T to -T and preserves the real part. Thus
S_D(v) is the real sum of paths from the three inner ports to the outer ports.
For each inner source attach a background counted by Z_v and grow the weighted
path tree in the punctured domain. Stop at an exit, a first self-collision, or
first contact with a background cycle. The initial total strength of the three
trees is 3 Z_v. A self-collision loop not enclosing v is approached from
outside and cancels with its reverse exactly as in the convex balance. A loop
enclosing v is approached from inside, since the inner source is in the same
component as v. Its corner at the collision vertex is reflex: the other loop
vertices contribute +-7 pi/3 of turning, and departure from the incoming stem
contributes +-pi/3. The two orientations therefore contribute
2 x^{|lambda|} cos((3/8)(8 pi/3)) = -2 x^{|lambda|},
which cancels first contact with the same loop added to the background. The
inverse operation, removing the first contacted (necessarily innermost)
background cycle and following it from the contact in either orientation, is
defined by first-contact without earlier stem intersection; every background-
contact term is canceled exactly once.
An outer exit can be reached with no background cycle at present, since the
source starts inside every such cycle; its real sum is the S_D(v) identified
above. An exit to another inner port, closed through v, gives a simple cycle
lambda through v, and every remaining background cycle surrounds it. If
L = |lambda|, opening the cycle at v leaves two directed inner-return paths
of weight x^{L-1} each. At a convex corner their turns are +-5 pi/3, at a
reflex corner +-7 pi/3: the full loop turn is +-2 pi with the turn at v
removed. Write t_lambda = 5 pi/3 or 7 pi/3. The remaining real tree identity
is
3 Z_v = S_D(v) + sum_{lambda ni v} 2 x^{|lambda| - 1}
cos((3/8) t_lambda) Z_out(lambda),
and moving the inner-return terms to the left supplies
-(2/x) cos(5 pi/8) = sqrt 2,
-(2/x) cos(7 pi/8) = 2 + sqrt 2,
which is the first displayed identity.
A cycle avoiding v places all three incident face centers in the same
component as v; a cycle through v encloses exactly one incident face center
at a convex corner and two at a reflex corner, and in a surrounding
configuration must be innermost (an inner cycle avoiding v would also enclose
v, lying on the outer cycle). This partitions the Z_D(f) configurations into
the three cases and proves the second displayed identity. Comparing coefficients
gives the (1/sqrt 2) two-sided visit comparison, and convex boundary positivity
gives the ordinary visit mass bound. QED
Flat-hexagon conclusions. Let Z(R) be the +2 nesting partition around the
central face in D_R and, for honeycomb v in D_R, define V_R(v) as the raw
chord mass through v. Then E_R^chord N = Z_R^chord^{-1} sum_v V_R(v). For
v at distance at most R/10 from the center, every first-exit boundary chord
through v is macroscopic for large R, so translating a hexagon of side
floor(R/2) centered at each incident face inside D_R gives
V_R(v) >= c Z(floor(R/2)),
uniformly for these bulk vertices. For v not adjacent to a boundary port,
a translated hexagon of side 3 R contains D_R, giving V_R(v) <= C Z(3 R)
uniformly. There are order R^2 bulk vertices and order R boundary-adjacent
vertices, each of which contributes at most one. Hence
c R^2 Z(floor(R/2)) / Z_R^chord <= E_R^chord N
<= C R^2 Z(3 R) / Z_R^chord + C R.
The chord partition Z_R^chord and the +2 nesting partition Z(R) will be
identified later: Z_R^chord = R^{3/4 + o(1)} and Z(R) = R^{1/12 + o(1)}.
Combining gives the mean E_R^chord N = R^{4/3 + o(1)} and bulk visit
probability R^{-2/3 + o(1)}. The proofs of the two identities occupy the
bulk of the paper.
The finite spin sum.
Place two groups A and B of m sites each on the horizontal cut of the
cylinder, m even. A spin configuration assigns each site a value in
{-1,0,1} and is retained only when the total charge in each group is zero.
A site carries a variable x_a and a chosen square root t_a, x_a = t_a^2.
The two-group spin sum F(X) is the sum over retained configurations of the
product of the two-site factors, multiplied by the square-root normalisation
prod_a t_a^{s_a^2}. Empty products have value one. The two-site factors
factorise, for each ordered pair of spin values, into two pieces: a
two-by-two Schur-type determinant contribution, antisymmetric in the pair,
and a symmetric Gaussian kernel. This factorisation is what makes the sum
reproducible by a Pfaffian contour integral below.
Individual terms have simple poles when two site variables coincide. The
sum is regular there. Residues cancel in pairs: a pair of sites with
mismatched occupancy can exchange which coordinate carries the nonzero
spin without changing the total charge content of the configuration, and the
two residue coefficients are opposite. Iterating the exchange on a group of
r coincident sites reduces the pole order by one at each step, so the sum is
regular at simultaneous coincidences of any multiplicity.
Contour replacement. For each fixed number of sites, the value of F at
homogeneous rapidities is replaced by a periodic Pfaffian contour
expression. Its limit as the horizontal period tends to infinity recovers
F. Concretely, write w_j for the logarithmic site coordinates and let P be
the period; the periodic contraction kernel is the sum over horizontal
images of a fixed odd kernel, and the Pfaffian of the resulting matrix
equals the finite spin sum term by term. Two inequalities control the
passage to the limit: the contraction kernel decays exponentially in the
imaginary direction, and the diagonal of the Pfaffian matrix is bounded by a
constant multiple of the off-diagonal mass. The limit is therefore uniform
on compact sets of site coordinates.
Two kernels occur. With tilde B_beta(t) = sinh t / (cosh t + cos beta),
put
B_0(w) = tilde B_{pi/4}(pi w / 4), C(w) = tilde B_{pi/2}(pi w / 6).
Both are odd and analytic on |Im w| < 3, both are strictly increasing on the
real line with B_0(w) > 0, C(w) > 0 for w > 0, and both are the ratios of an
odd kernel to its own divided-difference normalisation. Their derivative
transforms are
hat{B_0'}(k) = 8k cosh k / sinh 4k, hat{C'}(k) = 6k / sinh 3k,
using hat h(k) = int_R e^{-ikw} h(w) dw. Both densities are strictly positive
and even, all moments are finite, and
hat{B_0'}(k) / hat{C'}(k) = (2/3) e^{phi(k)},
phi(k) = log((1+t^2)(1-t^6)/(1-t^8)), t = e^{-|k|},
continued by phi(0) = log(3/2). The identity follows from
hat{tilde B_beta'}(xi) = 2 pi xi cosh(beta xi) / sinh(pi xi), obtained by
applying cos beta + sin beta partial_beta to the transform
int_R e^{i xi t} dt / (cosh t + cos beta) = 2 pi / (sinh(pi xi/2)) times
the corresponding factor, and then scaling.
Transfer matrix. Let V = C^3 with basis |0>, |+>, |->, let s act
diagonally with eigenvalues 0, +1, -1, and consider a row of M rhombi with
row parameters phi_j. Remove the telescoping phase exp(i theta (left -
right)/4), and set q = e^{2i alpha}. The oriented row matrix with twist p is
T_p(w) = Tr_a p^{s_a} R_{a1}(xi - phi_1) ... R_{aM}(xi - phi_M),
p in {i,-i}, w = e^{2i xi}, x_j = e^{2i phi_j}.
On the charge-zero subspace H_M of (C^3)^{tensor M}, the local limits of the
sine weights give
z = w/x_j -> 0: (a,b,v) -> (q^{-3}, q^3, 1), (u,d) = O(z^{1/2}),
z -> infinity: (a,b,v) -> (q^3, q^{-3}, 1), (u,d) = O(z^{-1/2}),
holomorphic in z^{1/2} and z^{-1/2} respectively. On a state with auxiliary
spin r and site spin s the diagonal limit is q^{3rs} at infinity and q^{-3rs}
at zero. The only surviving off-diagonal move at infinity changes the
auxiliary spin from -1 to 1 and the site spin from 1 to -1, and at zero the
reverse. A surviving move at an inactive site therefore preserves vacancy
parity. Square roots occur only in the single turns, whose number in an
auxiliary trace is even, so T_p is rational in w with possible simple poles
at q^5 x_j and q^6 x_j. Changing a square-root sign conjugates T_p by the
parity operator P_j = (-1)^{s_j^2}. At either endpoint of the row parameter
a closed auxiliary trace cannot use a strictly one-directional off-diagonal
move, and the remaining diagonal trace is
sum_{r=-1}^{1} p^r q^{+-3r sum_j s_j} = 1 on H_M. Hence
T_p(0) = T_p(infinity) = I.
With C_l(z,b) = z/(z - q^l b) this gives the partial-fraction form
H_p(w) = T_p(w) - I = sum_{j=1}^{M} sum_{t=0}^{1} C_{5+t}(w, x_j) U_{jt},
where the coefficient operators commute, the basis functions vanish at zero
and have distinct poles, and the condition at infinity additionally forces
sum_{j,t} U_{jt} = 0.
Fixed vacuum lines. Let A be a generic linear combination of the finitely
many probe deviations T_p(w_nu) - I, and let d_A be the scalar derivative
partial_rho det(rho I - A) at rho = 0. Then zero is an exact eigenvalue of
A, the projector is Pi_p = adj(-A) / d_A, and on the union of the three
scales every diagonal block with a charged spectator is invertible. With all
scales singleton and probes at the swap points, a diagonal block carries the
eigenvalue p_j^{S_j}; odd total charge forces the exponent of q in p_j to be
odd, so p_j^{S_j} is not 1, and only the all-empty block can carry the
simultaneous character, with multiplicity one. A generic combination of the
probe deviations therefore has a simple zero eigenvalue and, by rationality in
square-root coordinates, a generically simple one, with no Jordan extension.
Separated scales. Let positive scales rho_1, ..., rho_r satisfy
rho_{k+1}/rho_k -> infinity, and let x_i/rho_{k_i} tend to a finite nonzero
limit. With consistent local square roots and a probe w/rho_j tending to a
finite nonzero limit at which the active local matrices are regular, the
limiting row matrices are triangular in the grading sum_i k_i s_i with row
grade at least column grade. Their diagonal grade blocks preserve every
scale total S_k = sum_{i: k_i = k} s_i and act on the active scale j with
twist
p_j = p q^{3 sum_{k != j} sgn(j-k) S_k},
and each block entry is the corresponding active-transfer entry multiplied by
F(s^{row})/F(s^{col}), with
F(s) = q^{(3/2) sum_{l= 0, while total charge
conservation gives Delta(sum_i k_i s_i) = sum_i (k_i - j) Delta s_i >= 0,
strictly unless every inactive charge is unchanged. An inactive site
contributes the diagonal factor q^{3 sgn(j - k_i) h_i s_i} with incoming
horizontal spin h_i = h_0 + sum_{l, L_{-i} = <0| Pi_{-i}.
For M >= 1, generic nonzero X, and sigma in {-1,0,1}^M with zero total charge,
put
V_sigma^R(X) = N(X) (R_i)_sigma / prod_j t_j^{sigma_j^2},
V_sigma^L(X) = N(X) (L_{-i})_sigma / prod_j t_j^{sigma_j^2}.
The quotient by the normalizer N removes the common poles, and the functions
are independent of the square-root signs. They extend to polynomials in
C[x_1,...,x_M] with
deg_{x_j} V_sigma^epsilon <= M - 1 - sigma_j^2.
With X_hat j the deletion of coordinate j and sigma_hat the corresponding
deletion, for sigma_j = 0 and either superscript,
V_sigma(X)|_{x_j = 0} = (prod_{k != j} x_k) V_{sigma_hat}(X_hat j),
[x_j^{M-1}] V_sigma(X) = V_{sigma_hat}(X_hat j).
Proof. Site sign covariance gives Pi_p(-t_j) =
P_j Pi_p(t_j) P_j, and the empty vector has even vacancy parity, so
(R_p)_sigma(-t_j) = (-1)^{sigma_j^2} (R_p)_sigma(t_j) and likewise for L_p.
The determinant formula makes the quotients rational in the t_j and invariant
under every sign change. The invariant rational field is generated by the
squares, so both functions are rational in the x_j.
For the pole removal, work with widely separated site magnitudes. One
spectator on the same scale as the varied site forces every other site to be
empty in a diagonal block carrying the fixed character, the active pair then
has total zero and original twist p, and with coordinates u, v in
left-to-right order and z = u/v the probe w = u is the twisted swap
reduction R(phi_u - phi_v) diag(p^{s_first}). Its neutral block has
eigenvalues 1, a, -a with
a = q^3 (z - q^2)(z - q^3) / ((z - q^6)(z - q^5)),
the eigenvalue 1 coming from the vacuum character, trace 1, and trace of the
square 1 + 2a^2. Normalizing the right and left fixed vectors r, l at the
state (0,0), direct multiplication gives
r_- = i r_+, l_- = i l_+, l r = 1,
r_+ = d_tile lambda / (1 - i a / p), l_+ = d_tile p / (lambda (1 - i a p)),
where d_tile is the nonzero local single-turn weight. At a pole of a the
reverse probe w = v is regular with a(1/z) = 0, so with
D(z) = (z - q^5)(z - q^6) the crossings are explicit:
a - 1 = (q^3 - 1)(z^2 - 1) / D(z),
a + 1 = (q^3 + 1)(z^2 + 2(1 + sqrt 2) z + 1) / D(z).
At z = 1 use identical rapidity branches and also probe at w = q u; the
twisted swap then has one eigenvalue 1 and one eigenvalue
v_tile^2 + a_tile^2 + b_tile^2, which is not 1, so joint simplicity holds.
Uniformity. Take a sequence of generic spectator lists whose scale gaps
tend to infinity and a putative pole zeta not 0 or infinity. After passing to
a subsequence, zeta lies either on a separated singleton scale or in a pair
whose ratio tends to a finite nonzero limit, and the scale probes converge
locally analytically in a square-root chart. Joint simplicity holds in the
singleton case and at every simple pair ratio. The only excluded ratios are
-1 and the two negative roots of z^2 + 2(1 + sqrt 2) z + 1. Around one of
those, take a small complex disk and use the pair probe w = u together with
the spectator singleton probes. The singleton tests keep every block with a
charged spectator invertible, the pair eigenvalues cross transversely by the
displayed factorisations, and the limiting scalar derivative has a simple
zero at the central ratio. Convergence on the boundary of the disk together
with the argument principle gives exactly one zero counted with multiplicity
for large gaps, hence a unique simple zero; the projector has at most one
simple pole in the disk.
After dividing by the half powers the endpoint orders give no pole at zero,
and at infinity deg_{x_j} N <= M - 1 gives degree at most M - 1 -
sigma_j^2. The set of separated generic spectator values is Zariski dense,
a relatively prime numerator and denominator in x_j stay relatively prime for
generic specialisations of the spectators, so the one-variable conclusion
holds over their coefficient field. Applying this successively to every j
removes every irreducible denominator and proves joint polynomiality,
including at simultaneous divisor intersections. The deletion values follow
from the coordinate limits. QED
Scalar characterisation. Write e_k(u,v) = v - q^k u, put
F_{A,B} = N(X)^2 G(X; A, B) for the pairing G of the two vacuum families,
and record the four properties that determine it. The empty-list value is 1.
For M = |A| + |B| >= 1 each polynomial is symmetric within the two groups and
has individual degree at most 2(M-1). Deleting a site j gives
F_{A,B}|_{x_j = 0} = (prod_{k != j} x_k^2) F_{hat j},
[x_j^{2(M-1)}] F_{A,B} = F_{hat j}.
For two sites u, q^delta u in one group, delta = 2 or 3,
F_{A,B}(u, q^delta u, Z)
= (u + q^delta u)^2 (prod_{z in Z} ell_delta(u,z)^2) F_red(X_red),
where X_red is the reduced list with the pair replaced by one site and
ell_delta(u,z) = z^2 - (q^delta + q^{-delta}) u z + u^2 measures the shift.
When |A| = |B| the quotient of F by N^2 has the paired-scale limit at
generic fixed pair ratios. This family is the unique family with these
properties.
Proof of uniqueness. Induct on M and take the difference of two such
families. If the larger group has size n > M/2, vary one of its sites.
Induction gives zeros at the 4(n-1) fusion points q^{+-2}, q^{+-3} times the
other sites of that group and at zero, and the top coefficient vanishes, so
the degree is at most 2M - 3. Since 4(n-1) + 1 > 2M - 3, the difference is
zero.
If both groups have size n, then M = 2n, the vanished top coefficient reduces
the degree in each variable to at most 2M - 3, and the zero at the origin
together with the four zeros for each same-group partner gives degree
1 + 4(n-1) = 2M - 3. Divisibility by these distinct linear factors shows that
the difference is a constant in the site variables times
prod_i x_i prod_{same-group pairs i,j}
e_2(x_i,x_j) e_3(x_i,x_j) e_5(x_i,x_j) e_6(x_i,x_j).
To determine the constant, put matched coordinates u_j, v_j on successive
increasing scales rho_j. At generic ratios the normalizer has the limit
N(X) ~ prod_j (u_j + v_j) prod_{j 0 annihilate the vacuum. Let L_0 be the total creation
energy, put qq = e^{-2 pi / T}, and let U fix the vacuum and act by
Psi_j -> -Psi_{j-1} under conjugation. Set Gamma = qq^{L_0} U.
Conditional angular trace. Fix T > 0, m >= 0, theta in C, and a homogeneous
analytic sample of the Gaussian and opposite-uniform fields. The normalising
trace is
Z_Gamma = Tr Gamma = prod_{r in 1/2 + N_0} (1 - qq^{6r}) > 0.
For s >= 1, if the folded coordinates have strictly increasing imaginary
parts with total range less than 1, then
(-2 pi i / (a_* T))^s Tr(Gamma Psi_{j_1}(v_1) ... Psi_{j_{2s}}(v_{2s}))
/ Z_Gamma
= Pf_{1 <= a,b <= 2s} C^{[T]}(v_b - v_a + i (j_b - j_a)).
The trace denotes its absolutely convergent ordered mode sum; for s = 0 both
sides are 1; odd products vanish; the formula at other nonsingular
coordinates is its meromorphic continuation.
Proof. The operator qq^{L_0} is trace class because
prod_{r>0} (1 + qq^r)^6 < infinity, and U is unitary. At each positive energy
r the exterior trace is the determinant of the identity plus the one-particle
action, which is -qq^r times a six-cycle, so its determinant is 1 - qq^{6r};
the sum of qq^{6r} converges, giving the product and its positivity.
Cyclicity for bounded individual CAR modes gives, for r > 0,
C_{jl} := Tr(Gamma Psi_{j,r} Psi_{l,-r}) / Z_Gamma
= sum_{n>=0} qq^{rn} delta_{l-n, j*},
from the recurrence C_{jl} = delta_{l,j*} + qq^r C_{j+1,l}. The CAR then give
-sum_{n>=1} qq^{-rn} delta_{l+n,j*} for r < 0, and the other energy pairings
vanish. Summing the half-integer modes for 0 < Im(v-u) < 1 gives
Tr(Gamma Psi_j(u) Psi_l(v)) / Z_Gamma
= (i/2) sum_{n in Z, n = l-j-3 mod 6} 1/sin(pi (v-u+in)/T),
the two geometric sums converging on the two sides of the annulus.
Multiplication by -2 pi i/(a_* T) gives C^{[T]}(v-u+i(l-j)). Moving one mode
cyclically in a higher product, contracting it by the CAR, and solving the
same six-step recurrence gives the Pfaffian rule. For ordered heights
eta_a < eta_b, direct contractions decay from eta_b - eta_a and contractions
wrapping through Gamma decay from 1 - (eta_b - eta_a); both are positive in
the stated strip, which makes finite products of mode sums absolute. Graded
alternation gives the meromorphic continuation. QED
Graded current bound. Let P_N project onto the energy-N subspace,
N in (1/2) N_0. For a fixed finite word Y of exponentials of the currents
X_{bc}(f) with periodic coefficients analytic on a common strip of positive
width, there are C, c > 0 with
||P_{N'} Y P_N|| <= C e^{C sqrt(N+1)} e^{-c |N'-N|},
uniformly in bounded numerical multiples of the currents in the
exponentials. The energy-level trace with Gamma is therefore absolutely
convergent, as is every exponential series of scalar trace coefficients.
Four-current trace. For the periodic current monomial with inputs b, b^{-1},
b_+^{-1}, b_- and the fixed analytic sample,
C_sigma = (1/Z_Gamma) Tr_gr(Gamma e^{X_{23}(b_+^{-1})} e^{X_{21}(b)}
e^{-X_{12}(b^{-1})} e^{-X_{16}(b_-)}),
where Tr_gr(Gamma Y) = sum_N qq^N tr_{P_N F}(U P_N Y P_N) with F the fermionic
exterior space.
Fermion contraction identity. For X = x + i a and (Phi_1, Phi_2) =
(U, V) or (V, U), there are constants c_1, c, C > 0 with
|G_Phi(X)| <= c_1 a^{-3/2},
|G_Phi(X) - c_2 (2 pi / a)^{1/2} e^{-X^* X / (2a)}| <= C (b/a)^{3/2},
where c_2 involves Phi_1(0) Phi_2(0) and their first moments, and
X^* X = x^2 + (a - b)^2. The derivation separates the two theta groups:
with w = exp(i Theta) and h = w^2,
g_{Phi_1,0} = e^{i r X},
g_{0,Phi_2} = e^{i s X} / [2 sinh(pi (X + i beta)/T)] with beta = 3 or 4,
and each factor is pairing-symmetric. With
w^{-2} = e^{-i Theta} h^{-1/2} = e^{-i Theta} y^{1/2} / (1 + y) and q = e^{2 i
alpha}, the variable change q -> q e^{i pi/2} passes to diagonal theta
differences. The short-time expansion of h^{-u} is u-specific, and the terms
with u not in {n/2} vanish by pairing antisymmetry, leaving the displayed
estimate after the constant, linear, and quadratic exponents are gathered.
The conditional trace identities hold for each fixed T. Passing the Gaussian
expectation through the reorganized expansion uses the absolute estimate of
the next section, and the gas identity is stated at sufficiently large integer
periods to which that estimate applies.
Bosonization to a horizontal gas.
The vertical coordinate of the cylinder compresses into a horizontal circle
of length T. Each spin configuration of the finite sum becomes an ordered
list of Weyl currents at the real projections of the sites, with an
oscillator trace over the auxiliary direction and a common state carried by
the running charge.
Three-root form. Fold the six sheets of the fermionic trace around the
horizontal cut into three ordered roots 1, 2, 3. For a normalised current
W_b = W_{b_w} exp(A) with A of zero spatial mean and
b_w = b_0 y^{-ell}, the action on a ground-charge state |c> = |c_1, c_2, c_3>
is
W_{b_w} |c> = (-1)^{c_1 + c_1 c_2} b_0^{ell + c_2 - c_1}
|c_2 + ell, c_1 - ell, c_3>,
together with interchange of the oscillator labels 1 and 2.
Verification. In the three-root word the modes with r + s = ell form
commuting blocks (1+F)(1+E')(1+F) with
F = b_0^{-1} Psi_{4,r} Psi_{2,s}, E' = -b_0 Psi_{5,-s} Psi_{1,-r}.
At opposite signs of r and s the block fixes the vacuum, at both positive it
contributes E', at both negative it contributes F. Grouping creators by
species and reversing the second energy list leaves
W_{b_w}|0> = b_0^ell |ell, -ell, 0>. Conjugation sends
Psi_1 -> -b_0^{-1} y^ell Psi_2 and Psi_2 -> b_0 y^{-ell} Psi_1, so
increasing c_1 multiplies the ground coefficient by
(-1)^{1+c_2} b_0^{-1}, increasing c_2 by (-1)^{c_1} b_0, and c_3 is inert.
These ratios together with the vacuum value give the displayed action.
Rotation of the ground charges sends d -> (d_2, d_3, -d_1) with sign
(-1)^{sum d_j + d_1(d_2 + d_3)}, counting the creator signs and the move of
the first species list past the other two. Applying rotation to the action
above gives the seam-twisted form
Gamma W_{b_w} |c> = qq^{|c''|^2 / 2} (-1)^{c_2 + c_3 + c_2 c_3}
b_0^{ell + c_2 - c_1} |c''>, c'' = (c_1 - ell, c_3, -c_2 - ell),
with qq = e^{-2 pi / T}. On creation oscillators of frequency n > 0 it acts,
in addition to the ground action, by B_n = qq^n diag(1, R) with
R = [[0,1],[-1,0]].
Oscillator contractions. For normalised creators with coefficient vector u
followed by annihilators with vector v', the oscillator trace with
propagator B_n, divided by its empty trace, is
exp( v'^t B_n (I - B_n)^{-1} u ).
On one oscillator with multiplier t, cyclicity and the commutator give
<(a^*)^l a^l> = l! (t/(1-t))^l and zero for unbalanced moments;
diagonalising the unitary part gives the vector formula. Reordering two
exponentials adds the contraction of the earlier annihilator with the later
creator; an unsplit exponential adds half its internal contraction.
Applied to the zero-mean source exp(sum_n A_n (-h_{1,n} + h_{2,n})) with
k = 2 pi n / T > 0 and s = (-1, 1, 0)^t, the Fock identity gives
s^t ( (1/2) + B_n (I - B_n)^{-1} ) s
= (1/2) coth(k/2) + (1/2) tanh k
= cosh(3k/2) / (2 sinh(k/2) cosh k) =: E(k),
and with D(k) = k E(k)/(2 pi) the source self-contraction is
exp( (1/2) int_T A D A ). Cross contractions dress a positive short and a
positive outer field by
f_s^osc(w) = -sum_{k != 0} A_n e^{ik w} / (2 sinh(k/2)),
f_o^osc(w) = sum_{k != 0} A_n e^{ik w} / (2 cosh k),
a negative field receiving the negative exponent. In the short coordinate
y^n = qq^{n/2} e^{ik w}; the two outer coordinates carry y^n = qq^n e^{ik w}
and e^{ik w}. The empty oscillator trace is
D_W = prod_{n>=1} (1 - qq^n)^{-1} (1 + qq^{2n})^{-1}.
Signs. With N_j the number of fields of type j and n_j those at each root,
the Klein exponent apart from inversions is n_1(N_2 + N_3) + n_2 N_3, and
adding the seam sign and substituting the charge closure
n_1 = ell, n_2 = (F_3 - E_2 - ell)/2, n_3 = (-F_3 - E_2 - ell)/2 leaves,
modulo two,
(F_3 + E_2)/2 + #(3 before 2) + #(outer before short).
Each such phase is absorbed by the coordinate powers and the modular pair
factors; after the modular self factors remove (E_2+F_3)^2/4 and the pairs
contribute tanh(pi d'/4) with d' = w_l - w_i, the surviving sign is
(-1)^{#(outer before short)}, which combines with the odd like-family bases
to make the coordinate sum alternating under exchange of neighbouring
fields. Whole even current pairs may therefore be reordered without a sign,
and the total winding scalar per species is
exp( -pi T ell^2/12 - 2 pi ell^2/(3T) ).
Rationality and residues. Put Delta(X) = prod_{j x_j} R_{v_j v_k}(x_k + nT - x_j),
with the case j = k, n = 0 excluded and every j = k, n >= 1 included. The
constants gamma_v are the basic or fused activities, independently of the
species. For the empty configuration the particle and residual products are
one and the state heat factor remains.
First-contour identity. The conditional trace and oscillator calculation give
each fixed current coefficient, with two empty oscillator factors D_W/Z_Gamma
and two Poisson factors sqrt(T). Source removal gives D_g e^{m^2 kappa_T}.
The pointwise bound below justifies integrating and summing these
coefficients on the first contours. Signed Wick and constant-mode
integrations give the real heat completion, with prefactor
sqrt(8/9)/sqrt(2T), and cancel every winding-dependent exponential, leaving
D_ell^{-1}. The finite periodic contour identities give exactly the basic and
fused terms with the activities below; endpoint absolute bounds and the
fusion factorial justify summing them, and the signed algebra of the state
weight identifies the sum with Z_{m,T}. The eight equal full-tour
representatives supply 1/8.
Winding constant. With t = e^{-k},
lim_{T->infinity} kappa_T
= -2 int_0^inf t(1-t)(1-t^3)^2 / ((1+t^2)(1+t^3)(1+t^4)) dk/k
= 2 log( (2 (2 + sqrt 2)) / 9 ).
Evaluation. Multiplying the rational function R(t) of t by 1 - t^{24} gives
coefficients d_j = d_{24-j} for 1 <= j <= 23, whose first twelve values are
(1, -1, -1, -2, 3, 3, 1, -4, -3, -1, 3, 2),
and whose total after reflection is zero. The grouped partial sum through
period N is R(e^{-k})(1 - e^{-24(N+1)k}), whose quotient by k is dominated by
R(e^{-k})/k, integrable because R(e^{-k}) = O(k^3) at zero and O(e^{-k}) at
infinity; dominated convergence justifies the period expansion. Frullani's
log difference on each block makes its integral -sum_j d_j log(24n + j).
Pairing j with 24 - j, with exponent d_{12}/2 at j = 12, and using
prod_{n=0}^N (n+a)(n+1-a) / (N! (N+1)!) -> a prod_{n>=1}(1 - a^2/n^2)
= sin(pi a)/pi,
the exponentiated sum of paired logarithms becomes prod_{j=1}^{11}
sin(pi j/24)^{d_j}, and the sine product follows from finite odd sine
multiplication in the limit N' -> infinity. Collecting the resulting log
gives 2 log(2(2+sqrt2)/9).
Signed height measures and the coercive matrix.
Write mu_i^s, mu_i^o, mu_i^H for the signed measures on real heights in the
short, outer, and H columns of the particle table, for a label i in one
species. For example mu_P^s = delta_{1/2}, mu_P^o = delta_1,
mu_P^H = delta_0. The separate coefficient sign in the particle table belongs
to the local factor and does not multiply the measures. The fused measures
are the union of the U and L measures at zero displacement:
mu_F^s = -2 delta_0,
mu_F^o = delta_{1/2} - delta_{-1/2},
mu_F^H = -delta_{3/2} - delta_{-3/2}.
In the order P, M, O_+, O_-, U, L, F_pair the totals of both the short
measure and the H-measure give
bar q = (1, -1, 0, 0, -1, -1, -2),
so the physical charge vector is q_{(alpha,i)} = bar q_i e_alpha for species
alpha. For a label iota = (alpha, i) put c_iota = bar q_i/sqrt2,
c = (c_iota)_iota, a_c = 8/9. The jump of the common charge at label iota is
-c_iota.
Green kernels. For k > 0 the short and outer height kernels are
p(a) = cosh((1/2 - |a|) k) / (2 sinh(k/2)), |a| <= 1, period 1,
o(a) = sinh((1 - |a|) k) / (2 cosh k), |a| <= 2, antiperiod 2.
Both are positive covariance kernels, with Green series
p(a) = sum_{n in Z} k e^{2 pi i n a} / (k^2 + (2 pi n)^2),
o(a) = (1/2) sum_{n in Z} k e^{pi i (n+1/2) a} / (k^2 + pi^2 (n+1/2)^2).
Strict positivity is what makes the quadratic forms below positive definite
whenever their signed measures do not vanish identically. The Fourier
exponentials of the extreme H charges are
h_+(k) = (1/9) / k + O(k), h_-(k) = 1/k + O(k) (k -> 0),
h_+(k) = O(e^{-4k}), h_-(k) = 4 e^{-3k} + O(e^{-4k}) (k -> inf).
Set C_i(k) = int cosh(k a) d mu_i^H(a) and S_i(k) = int sinh(k a)
d mu_i^H(a), so C_i and S_i are the even and odd Fourier exponentials of
the H-measures. Define the dimensionless matrix H by
H_{alpha i, beta j}(k) = delta_{alpha beta}
[ int int p(a-b) d mu_i^s(a) d mu_j^s(b)
+ int int o(a-b) d mu_i^o(a) d mu_j^o(b) ]
+ sum_{sigma = +-} (P_sigma)_{alpha beta} Phi^sigma_{ij}(k),
with
Phi^{sigma, pre}_{ij} = -h_sigma C_i C_j,
Phi^{sigma, post}_{ij} = -h_sigma (C_i C_j - S_i S_j)
+ 4 * 1_{sigma = -} int int sinh(k (|a-b| - 3)_+)
d mu_i^H(a) d mu_j^H(b).
The pre-collision formula is the Gaussian absolute moment: reality of the
field on the real line gives Re xi(x+ia) = (xi(x+ia) + xi(x-ia))/2, which
replaces each signed charge by two signed halves without changing the sign of
its magnitude. The post-collision formula uses the original charges together
with cosh(k(a-b)) = cosh(ka)cosh(kb) - sinh(ka)sinh(kb), and the last term is
the continued half of the two extreme H-charges, nonzero only in the minus
channel and only after the collision.
Normalised coordinates. In this calculation only, replace F_pair by
F = F_pair/2, halving all three signed measures. The auxiliary coordinate can
be included in the pre-collision calculation and removed afterwards. Put
D_0 = diag(1, 1, 1, 1, 1, 1, 2), D_F = diag(D_0, D_0),
on P, M, O_+, O_-, U, L, F in each species. Then
H_phys = D_F H_norm D_F, and a physical coefficient vector v_phys has
normalised coordinates D_F v_phys, so the physical particle count is
represented by
n_norm = D_0^{-1} 1 = (1, 1, 1, 1, 1, 1, 1/2),
and the normalised charge row is (1, -1, 0, 0, -1, -1, -1). The fused label
F therefore has count 1/2 and charge -1, while R_d = (U+L)/2 has count 1 and
charge -1.
This difference is why the displacement must stay fixed. At d = 0 the height
measures of R_d and F agree but their count values do not, so the count bound
cannot be uniform through d = 0. The same-species U, L factor also carries
the simple collision pole of the boson section, with magnitude proportional
to 1/|x +- 2 i d| near collision, so pointwise constants depend on d. The
contour argument sums the series at fixed endpoints on the two sides of the
collision, which is precisely the fixed-displacement hypothesis.
Even core. In the quotient, the even direction (O_+ + O_-)/2 equals
(P+M)/2. On P, M, F the short and outer measures give
A_sigma = M_0 - h_sigma X X^t - sigma lambda V V^t,
M_0 = [[f+g, -f+g, -l+j_0],
[-f+g, f+g, l+j_0],
[-l+j_0, l+j_0, f+g/2]],
with
f = p(0), g = o(0), l = p(1/2), j_0 = o(1/2),
X = (1, 1 - 2 cosh k, -cosh(3k/2))^t, V = (0, 1, 1)^t.
For the third row, the normalised F short is -delta_0 and its two outers have
weights 1/2 and -1/2. The three even H-moments give X. The same core
submatrix appears for nonzero d. Strict Green positivity gives M_0 > 0:
cancelling its short measure forces equal P and M coefficients and a zero F
coefficient, and the remaining outer measure then forces both coefficients to
vanish.
Coercivity inequality. Let u = e^{-k/2} and set
x = X^t M_0^{-1} X, y = X^t M_0^{-1} V, v = V^t M_0^{-1} V.
Direct inversion gives
x = (1-u^2)(u^4+1)(u^8+u^6-u^4+u^2+1) / (3 u^6 (u^2+1)),
y = 2(u^2-1)(u^4+1) / (3 u^3),
v = (7u^2 - 12u + 7) / (3(1-u^2)).
The signs of 1 - h_+ x and 1 - h_- x are respectively positive and negative.
After positive denominators are cleared the numerators are
(u^4+1)(3u^8 + 2u^6 + 2u^4 + 2u^2 + 3)
and
-(1-u^2)^2 (u^4+1)(u^8 + 2u^6 - 2u^4 + 2u^2 + 1),
and the last parenthesis is positive because with v_0 = u^2 it equals
(1-v_0^2)^2 + 2 v_0 (1+v_0^2). Put
w_sigma = v + h_sigma y^2 / (1 - h_sigma x).
Substituting the rational formulas for h_sigma at t = u^2 and
lambda = 3(1-u^8)/(8(1+u^8)) gives
(1/w_sigma) - sigma lambda = ((1-u^2)/u) (N_sigma(z)/D_sigma(z)) > 0,
z = u + u^{-1} - 2,
with coefficients in ascending order
N_+ : (96, 1280, 5496, 9312, 7696, 3360, 772, 84, 3),
D_+ : (128, 2240, 13888, 38304, 55392, 45584, 21984, 6128, 912, 56),
N_- : (32, 256, 424, 1024, 1384, 832, 232, 28, 1),
D_- : (128, 2240, 13760, 36704, 49120, 36176, 15264, 3664, 464, 24).
All coefficients of N_+, D_+, N_-, D_- are positive for z >= 0, so the
right side is finite and positive, which prevents w_sigma from vanishing. In
the plus channel M_0 - h_+ X X^t > 0, so w_+ > 0 and the displayed inequality
is 1 - lambda w_+ > 0; both negative rank-one updates then preserve
positivity. In the minus channel the first update has exactly one negative
direction, and since w_- -> -3 as u -> 0 it is negative throughout, so the
inequality gives 1 + lambda w_- < 0. The determinant lemma makes the
determinant positive after adding lambda V V^t, and a positive rank-one
update cannot add negative directions, so it removes the unique negative
direction. Hence A_sigma > 0 for every k > 0.
The displaced and odd directions, and the absolute estimate.
The remaining even direction is R_d = (U+L)/2; put Z_d = R_d - F and
D_* = |d| k. The H-moment of R_d is -cosh((3/2+d)k). The odd directions are
O_odd = (O_+ - O_-)/2, W_d = (U - L)/2,
whose H-sinh moments are -sinh k and -sinh((3/2+d)k). Write T for the
continued correction in the post-collision minus channel. Summing the
opposite extreme charges with their actual half weights gives exactly
T_{R_d R_d} = 2 sinh(2D_*), T_{R_d F} = 2 sinh D_*,
T_{W_d W_d} = -2 sinh(2D_*),
and no other continued entries. These terms are absent before collision.
For small D_* and small |d| the square of Z_d is D_*(1 + O(D_* + |d|)) and
its crosses with the core are O(D_*), uniformly. For the field part the
short square is exactly
(3/2) p(0) + (1/2) p(2d) - 2 p(d)
= f(cosh D_* - 1)^2 + sinh D_* - (1/4) sinh(2D_*),
and the outer square is
o(0) - o(d) + o(1+d) - (1/2) o(1+2d).
Each has leading term D_*/2, and the Green formulas also give the stated
cross bounds. At small k the possibly larger second-order bounds
O(D_*^2/k) are still O(|d| D_*). The uncontinued H-square difference is
O(D_*^2) with crosses O(D_*). The continued contribution to the Z_d square
is 2 sinh(2D_*) - 4 sinh D_* = O(D_*^3), and the V-terms obey the same
bounds. The Schur cost of all crosses is at most
C D_*^2 / min(k,1) <= C max(|d|, D_*) D_*
for small D_* and |d|, because D_*/min(k,1) <= max(|d|, D_*). Absorption
against the core gap leaves a positive multiple of the D_* square and part
of the plus charge gap. The direction Z_d has zero total charge.
For the odd directions the outer part alone is
[[o(0), o(1/2+d)], [o(1/2+d), (o(0)+o(1+2d))/2]]
>= c_* min(k,1) I
for small |d|, by strict Green positivity and its limits at zero and
infinity, and the short part is nonnegative. The pre-collision H-part
vanishes. In the post-collision case its uncontinued part is the positive
rank-one form of the two sinh moments, and the only adverse continuation is
the last entry of the T-table. On bounded frequency intervals its cost is
O(|d| k); at large k its sum with the leading height diagonal is
e^{2D_*} - 2 sinh(2D_*) = e^{-2D_*},
up to exponentially small errors, and the height cross is exponentially
small. Choose a large-frequency cutoff absorbing these errors uniformly for
|d| in a small fixed interval, then choose |d| small enough on the remaining
bounded interval. This proves the odd gap.
Physical limit. The sector gaps show that the only null directions are the
two O-relations. At small k the common channel supplies the c c^t/k term and
every quotient direction has a k-order gap. After physical congruence the
same assertions hold with the physical count and charge, and the expansions
of p, o, h_sigma, the continued term and K give
H'(k) = (a_c/k) c c^t + k B_d + O_d(k^2)
for a real symmetric B_d, recording the exact leading coefficient that
cancels the state square. For fixed nonzero d, the high-frequency formulas
give exponential convergence to a positive quotient matrix. In particular,
for each fixed displacement in the small range, the physical H' has null
space the two O-relations for every k > 0, is bounded below on the quotient
by a positive multiple of k I + c c^t/k, and converges exponentially at
infinity to a strictly positive matrix.
Energy identity. Index the physical particles by alpha = 1, ..., N, with
labels iota_alpha and positions x_alpha in one period, and set
rho_iota = sum_{alpha: iota_alpha = iota} delta_{x_alpha},
hat rho(k) = int_T e^{-i k x} rho(dx), k in (2 pi / T) Z.
The forward regular energy, including nonzero self-images, is
E_R = sum_{alpha,beta} sum_{r in Z, x_beta + rT > x_alpha}
R_{iota_alpha iota_beta}(x_beta - x_alpha + rT)
= (1/2) sum_{alpha,beta} (R_T)_{iota_alpha iota_beta}
(x_beta - x_alpha)
- (1/2) sum_alpha R_{iota_alpha iota_alpha}(0)
= (1/(2T)) sum_{k in (2 pi/T) Z} hat rho(k)^* hat R(k) hat rho(k)
- (1/2) sum_alpha R_{iota_alpha iota_alpha}(0).
The half-double sum pairs the forward image (alpha, beta, r) with its
reverse (beta, alpha, -r). Only (alpha, alpha, 0) is excluded from the gas,
so the subtracted diagonal is the unperiodised value R(0) and every nonzero
self-image remains.
Young smoothing. Let g_T be the normalised periodic Gaussian with Fourier
multiplier e^{-k^2/2}, and let nu = n^t rho = sum_alpha delta_{x_alpha}.
No positivity of hat R(0) on vectors other than c is required: the mean
square in the energy bounds a positive multiple of T p_c^2 up to
O_theta(T), and the diagonal correction costs O_d(N) independently of T.
Combining the energy identity with the quotiented coercivity and including
the local factors gives, for every nonzero post-collision weight,
-log|W_{d,theta}(Y,p)|
>= c_2 ( ||g_T * nu||_{L^2(T)}^2 + ||g_T * p_c||_{L^2(T)}^2 )
- C_2 (N + T),
with c_2, C_2 depending only on d and Theta. The local constants have
logarithmic cost O_d(N), and the Berry factors carrying real state charges
are phases whose theta-dependent parts cost O_Theta(N). For the
pre-collision product the same identity holds with the real angle
vartheta = Re theta; the symmetrised local constants are finite for fixed
d < 0 and cost O_d(N), and the same bound holds with
-log W^{pre}_{d, Re theta} on the left.
Box counts. With h_T = g_T * g_T,
||g_T * nu||_2^2 = sum_{alpha,beta} h_T(x_alpha - x_beta)
>= c_1 sum_{j=0}^{T-1} n_j^2,
because h_T is positive with a uniform positive lower bound for pairs in the
same unit box. Each jump of p_c has magnitude at most sqrt 2. For x in box j
write p_c(x) - (g_T * p_c)(x) by integrating its increments against the
Gaussian; a jump in a box r boxes away is multiplied by a Gaussian tail
bounded by C e^{-c (r-2)_+^2}, and the resulting sequence of bounds is
summable uniformly after periodisation. Discrete Young's inequality therefore
gives
||p_c - g_T * p_c||_2^2 <= C_1 sum_j n_j^2.
Within box j the same jump bound gives
sum_j p_{c,j}^2 <= 2 ||p_c||_2^2 + C_2 sum_j n_j^2.
Using a smaller positive constant, the two smoothed norms consequently
control sum_j (n_j^2 + p_{c,j}^2). Finally
sum_j n_j <= eps sum_j n_j^2 + C_eps T absorbs the linear costs.
Absolute stability. There is 0 < d_0 < 1/2 with the following property. Fix a
compact set Theta in C and either a pre-collision displacement
-d_0 < d < 0 or a post-collision displacement 0 < d < d_0. There are
c > 0 and C < infinity, depending on d and Theta, such that for every
positive integer T and every theta in Theta, every finite configuration Y of
distinct real positions on R/TZ with the six basic labels, or with those
labels together with F_pair in the post-collision case, and every compatible
state path p with jumps -q_i and closure in Z^2/8Z(1,-1), satisfies
|W_{d,theta}(Y,p)| <= exp( C T - c sum_{j=0}^{T-1} (n_j^2 + p_{c,j}^2) )
in the post-collision case, with n_j the number of physical particles in
[j, j+1) and p_{c,j} = (p_1(j+) + p_2(j+))/sqrt 2. The same bound holds for
W^{pre}_{d, Re theta}(Y,p) in the pre-collision case. Zero weights already
satisfy the inequality.
Summability. For a fixed common integer p_1 + p_2 there are eight classes in
Z^2/8Z(1,-1), and the Gaussian bound on the common charge sums over that
integer. In each box, integration and the ordinary label factorials reduce
to a convergent sum of the form
sum_{n>=0} A^n e^{-c n^2} / n! < infinity,
and dropping configuration constraints only enlarges the bound. Both the
pre-collision Gaussian integration and the post-collision gas series are
therefore absolutely summable at the fixed displacements, locally uniformly
for theta in compact sets. The constants may depend on d and assert nothing
uniform as d approaches zero.
Compact propagation with Jordan blocks.
The horizontal gas has signed weights, so the partition is not the trace of a
positive operator. What is available is a compact Hilbert-Schmidt operator
whose power traces reproduce the gas, together with a gauge that turns the
signed local factors into a contraction.
Gauge. A real measurable function Phi on the state space exists with
|e^{Phi(S') - Phi(S)} w_theta(S,S')|
<= exp( C_b - delta_b (|Y_prev|^2 + |Y|^2 + p_c^2) ),
delta_b = c''/b > 0,
for theta in the compact parameter set and every allowed successor. To build
it, subtract a smaller quadratic cost from the edge costs used in the cycle
cost and add C_b, so that every finite directed cycle has nonnegative
modified cost. Let Phi(S) be the infimum of modified costs of paths from a
reference state to S. Every state can be reached and returned from: insert P
and M charges as needed at distinct positions, then insert the required final
block. A fixed return path bounds the infimum below by the negative of its
cost, so the infimum is finite, and the directed triangle inequality gives the
displayed bound. For measurability, restrict the intermediate positions in
the infimum to rational translated coordinates and use continuity on the open
simplices; the terminal state stays arbitrary and fixed. The discrete
constraints depend only on the labels, so on each label stratum the infimum is
countable and continuous in the endpoint, hence measurable.
The gauged kernels then define a Hilbert-Schmidt operator E = E(theta) on the
sequential Hilbert space, acting from successor to predecessor, with spectral
radius r_E < 1 in the parameter range and with peripheral spectral structure
described below. The Bargmann cycle identity supplies exactly the far
interactions and the local weights, giving the operator-gas trace identity
tr E^n = sum over configurations of n blocks of the gas weight.
Site tail. The deterministic site factors are omega_v(x)^m, and each omega_v
has modulus below 1 off the origin and tends to 1 at infinity with an
exponential correction of rate pi/3. Write kappa_1 b for that rate in blocks.
On an interval of length J = floor(log m /(kappa_1 b)) about the origin every
suppressed type therefore disappears; outside that interval the exterior
operator still allows all types. Over the inner interval the propagation
multiplier is r_I = 1, and over the exterior it is r_E, so the propagation of
the physical partition over n blocks carries the factor
exp( -(2 log m /(pi b/3)) (pi b/16) ) = m^{-3/8}
together with other bounded factors.
Empty-block traces. Let E_0 be the homogeneous operator with every site
multiplier equal to one, and let I_0 retain only the labels M and F_pair.
Both depend on theta. For n >= 2,
tr E_0^n = det(I - A_0^n)^{-1} / c(bn),
tr I_0^n = 8 det(I - A_0^n)^{-1} sum_{k in Z} u_k^n,
u_k = exp( -(4 pi b/9)(k + theta/pi)^2 ).
The first equality is the empty contour at m = 0 together with the gas
identity. In the second, every retained particle has strictly negative common
charge, so a closed sequence contains no particles, and the eight
representatives at each k = p_1 + p_2 give the displayed sum. The same
reasoning gives the trace formula with the sum restricted to a consecutive
range of k for a triangular diagonal compression to that range.
Spectral facts. For a compact Hilbert-Schmidt operator the limsup of the
absolute n-th roots of the power traces is the spectral radius, and those
traces determine every nonzero eigenvalue with its algebraic multiplicity.
Approximate the operator by finite rank with norm error below eps. Outside
radius eps the error resolvent exists, and finite-dimensional inversion makes
the full resolvent meromorphic with finite-rank pole parts. Resolvent contour
integration isolates the Riesz blocks, the resolvent identity makes each
Riesz projection idempotent and commuting, and Laurent comparison identifies
the higher pole parts with powers of the nilpotent part on that block. The
complementary powers are exponentially bounded at any radius containing the
remaining spectrum, and their traces have the same rate by retaining two
Hilbert-Schmidt factors. On an isolated spectral radius the trace is a sum of
algebraic multiplicities times eigenvalue powers, and Cesaro extraction of
distinct unit-circle frequencies shows that such a sum cannot tend to zero
unless all coefficients vanish. Applying this successively off zero proves
the assertion, and separating resolvent contours give locally uniform bounds
for norm-continuous families.
Bare trace expansion. Compactness makes the peripheral set P = {zeta in
sigma(E): |zeta| = 1} finite and nonempty. On its finite Riesz blocks,
E^n P_zeta = zeta^n sum_{t=0}^{q_zeta - 1} binom(n, t) zeta^{-t}
J_zeta^t P_zeta,
J_zeta = (E - zeta I) P_zeta, J_zeta^{q_zeta} = 0,
and the complementary restriction E_< has spectral radius strictly below one.
The remainder is O(rho^n) in operator norm for some rho < 1, and
||E^n|| <= K_0 (1+n)^q for a fixed q. For n >= 2 the complementary trace norm
is bounded by ||E_<||_2^2 ||E_<^{n-2}||. Every positive nilpotent power has
trace zero on a finite Riesz block, so
tr E^n = sum_{zeta in P} d_zeta zeta^n + O(rho_1^n), rho_1 < 1.
Coefficient polynomial. Index the two exterior gaps by L, expand the exterior
factors, and rotate the trace to begin with C'_m = (u/r_E)^{2L} C~_m. The
term for two compatible exterior words has the form
tr( C'_m W_+ (E')^{s - l_+ - l_-} W_- ), s = n - 2L.
For a peripheral eigenvalue zeta of E' write P_zeta for its Riesz projection,
d_zeta for the rank, and J_zeta = (E' - zeta) P_zeta. The coefficient
polynomial supplied by the trace lemma is
P_{zeta,m}(s) = sum_{W_+,W_-} zeta^{-l_+ - l_-}
tr( C'_m W_+ [ sum_{t>=0} binom(s - l_+ - l_-, t) zeta^{-t}
J_zeta^t P_zeta ] W_- ).
The nilpotent sum is finite, and the word sum is coefficientwise absolutely
convergent by the exterior word bound. The lemma then gives
R_m = P_{zeta,m}(0) / (d_zeta zeta^{2L}), |R_m| <= C |u/r_E|^{2L},
with the constant uniform in m on the fixed parameter neighbourhood, since the
word moments, the central trace norm and the peripheral nilpotent bounds are
uniform and |zeta^{2L}| = 1 makes the changing offset harmless.
Two-gap coefficient lemma. Keep two distinct frequencies zeta, eta with
expansions of order h_z, h_e. For a word sum over W_1, W_2, W_3, W_4 with
exterior lengths l_+ and l_-, set
P~_{zeta,eta}(z) = sum_{W} zeta^{-l_+} eta^{-l_-}
tr( C W_1 B_zeta^+(z, l_+) W_2 W_3 B_eta^-(z, l_-) W_4 ).
Then for every frequency beta in {alpha^4: alpha in P_+} union
{(zeta eta)^2: zeta in P_+, eta in P_-},
R sum_{alpha in P_+, alpha^4 = beta} rank P_alpha^+
= sum_{zeta in P_+, eta in P_-, (zeta eta)^2 = beta}
(zeta eta)^{-L} P~_{zeta,eta}(-L),
and
|R| <= K (1+L)^h ||C||_1, h = max_{zeta,eta}(q_zeta^+ + q_eta^- - 2),
with K depending only on E_+, E_-, the parameter neighbourhood and the gauge,
and independent of C, L and R.
Signed trace coefficients. Let E be a Hilbert-Schmidt operator on a complex
Hilbert space with spectral radius one, P = {zeta in sigma(E): |zeta| = 1},
Riesz projections P_zeta of rank d_zeta, and J_zeta = (E - zeta I) P_zeta with
J_zeta^{q_zeta} = 0 on the range. Let C be trace class. Suppose two countable
families W_+, W_- of bounded operators have nonnegative integer lengths
l(W_+-) with
sum_{W_+-} e^{eps l(W_+-)} ||W_+-|| < infinity
for some eps > 0. For integers s >= 0 put
F_s = sum_{l(W_+) + l(W_-) <= s}
tr( C W_+ E^{s - l(W_+) - l(W_-)} W_- ),
and assume a >= 0 and R in C satisfy F_s - R tr E^{s+a} -> 0 as s tends to
infinity, using only s with s + a >= 2. Then the coefficientwise absolutely
convergent polynomial
Q_zeta(z) = sum_{W_+,W_-} zeta^{-l}
tr( C W_+ [ sum_{t=0}^{q_zeta - 1} binom(z - l, t) zeta^{-t}
J_zeta^t P_zeta ] W_- ), l = l(W_+) + l(W_-),
is identically R d_zeta zeta^a. In particular
R = Q_zeta(0) / (d_zeta zeta^a), |R| <= K ||C||_1,
with K depending on E and the bounds in the moment condition but not on C or
a; more precisely K may be chosen from eps, the two weighted word sums, the
finite nilpotent orders q_zeta and the norms ||J_zeta^t P_zeta||. The
conclusion is uniform for a family with uniform bounds on those data, and no
peripheral semisimplicity is assumed.
Proof. The power expansion of E on a peripheral block and the geometric
integration of the binomial polynomials give the identity for every finite
word sum. Concretely, for integer s >= 0 the block expansion reads
sum_{W} zeta^{-l} tr( C W_+ sum_{t=0}^{q-1} binom(s - l, t) zeta^{-t}
J^t P W_- )
= zeta^{-s} F_s
with F_s the truncated trace, because the binomial coefficient
binom(s - l, t) is the number of ways to distribute t nilpotent insertions
among s - l slots and the word sum reproduces the expansion coefficients of
E^{s-l}. The right side differs from R zeta^a d_zeta by a term tending to zero
in s, and the left side, being a polynomial in s of degree at most q - 1, must
equal that constant identically once the difference tends to zero on
infinitely many integers. Evaluating at s = 0 gives the displayed formula for
R. The trace-class norm of each word term is bounded by
K_1 ||C||_1 ||W_+|| ||W_-|| (1 + l)^{q_zeta - 1}, whose sum is finite by the
moment condition, and since d_zeta >= 1 and |zeta^a| = 1 the stated bound
follows. QED