At the critical weight a self-avoiding path on the honeycomb lattice can
travel a long distance without exponential cost. How much of its total
weight reaches the far side of a strip is a sharp question. For a strip
N bands high the crossing mass is comparable to N^(-1/4), the first
horizontal displacement moment of returning paths is comparable to
N^(3/4), and an exact conservation law ties the two. The multiplicative
constants are absolute.
The honeycomb lattice is the dual of the equilateral triangular tiling of
side one, one edge direction horizontal. A port is the midpoint of an edge
on the boundary of a region made of triangles. A path between two distinct
ports follows dual edges, includes the two terminal half-edges, visits each
dual vertex at most once and meets the boundary only at its two ends. Its
length is the number of visited dual vertices; ports carry no weight. The
path has weight rho^length with
rho = 1/sqrt(2 + sqrt 2) = 1/(2 cos(pi/8)).
The reciprocal is the honeycomb connective constant, shown equal to
sqrt(2 + sqrt 2) by Duminil-Copin and Smirnov. Thus rho is the largest
weight for which the path sums below converge.
https://arxiv.org/pdf/1007.0575
Let S_N be the infinite horizontal strip of N triangular bands.
Consecutive band boundaries are sqrt(3)/2 apart. Index the bottom ports by
the integers, consecutive ports one horizontal unit apart, and put the
source at bottom port 0. For ports a, b let Z_D(a,b) be the sum of
rho^length over admissible paths from a to b inside a region D. An arch
returns to the bottom boundary and a bridge ends on the top boundary; all
intermediate vertices lie strictly inside the strip. Set
K_N(k) = Z_{S_N}(0,k),
A_N = sum over k != 0 of K_N(k),
B_N = sum over top ports b of Z_{S_N}(0,b),
m_N = sum over k >= 1 of k K_N(k).
Top ports lie on N/2 + Z, and these are unnormalised masses. The main
result is the following.
For every integer N >= 1 all four sums are finite and, with constants
0 < a < b < infinity independent of N,
c A_N + B_N = 1, c = cos(3 pi/8),
a B_N <= m_{N+1} - m_N <= b B_N,
a N^(3/4) <= m_N <= b N^(3/4),
a N^(-1/4) <= B_N <= b N^(-1/4),
and B_N is nonincreasing in N.
The exponent 3/4 shadows the planar boundary exponent 5/8. Summing the
far endpoint along a side of the strip turns -5/4 into -1/4, and a first
moment turns -1/4 into 3/4. The theorem pins constants, not only the
exponent. Nienhuis predicted the honeycomb critical point and the
self-avoiding exponents from the O(n) Coulomb gas.
https://doi.org/10.1103/PhysRevLett.49.1062
Lawler, Schramm and Werner supplied the conformally invariant scaling
picture, with boundary exponent 5/8, predicting the -5/4 boundary mass and
hence the -1/4 strip power.
https://arxiv.org/pdf/math/0204277
Duminil-Copin and Smirnov proved the critical point by a parafermionic
observable whose local cancellation yields a positive boundary identity,
yet their strip bounds, of order 1/N and 1, leave the power open. Beaton,
Bousquet-Melou, de Gier, Duminil-Copin and Guttmann proved decay of the
crossing mass through surface adsorption.
https://doi.org/10.1007/s00220-014-1896-1
Glazman and Manolescu gave a shorter decay proof and invariance of the
half-plane boundary two-point function under rhombic deformations, and
Krachun and Panagiotis proved a polynomial upper bound together with
quantitative sub-ballisticity.
https://doi.org/10.1214/19-AIHP1024
https://arxiv.org/pdf/2310.17299
The route computes m_N first and recovers B_N from the increments
m_{N+1} - m_N. It passes from finite-width strip transfers to a polynomial
formula, then to a positive integral estimated uniformly in the height.
Throughout write
t = 3/8, lambda = pi/8, d = cos(2 lambda) = sin(2 lambda),
C = cos(lambda), c = cos(3 lambda), eta = 2d - 1.
Then rho = C - c = 1/(2C), and c = eta C, C = eta(2C + c). A single turn
of a honeycomb path is +-pi/3 and t(pi/3) = pi/8 = lambda, so weighting a
turn by e^(i lambda) or e^(-i lambda) is the same as weighting the total
signed turning by e^(i t W).
The first ingredient is a conservation law.
Boundary identity. Let D be a finite simply connected union of triangles
with simple polygonal boundary and a a boundary port. For a path gamma from
a let W(gamma) be its total signed turning, from the inward initial
direction to the outward final direction. Then
sum over boundary ports b sum over paths gamma from a to b in D
rho^length(gamma) e^(i t W(gamma)) = 1.
If D is convex, the unsigned boundary sum, taken over all paths from a to
any boundary port, is at most 1/c.
Proof. Send mass one out of a along the inward half-edge. At each newly
reached interior vertex split the mass between the two allowed
continuations, multiplying by rho e^(i lambda) and rho e^(-i lambda). The
relation 2 rho cos(lambda) = 1 conserves total mass at every split.
Stop a branch on exit through a boundary port, or when the next step would
enter a vertex the branch already visited. The first case is terminal and
gives the paths in the sum. The second closes a simple cycle. The newly
closed cycle is simple and the incoming stem is disjoint from it, since the
stem leaves the boundary and cannot cross the cycle before their first
common vertex. The unused branch at the collision vertex is therefore
outside the cycle, and the cycle turns there by +pi/3 rather than -pi/3.
The turn from the stem into the cycle reverses the sign of that corner, so
the two traversals of the cycle carry winding increments +4pi/3 and
-4pi/3 relative to the stem. They have equal unsigned weight and cancel,
because e^(i t 4pi/3) + e^(-i t 4pi/3) = 2 cos(pi/2) = 0. Cycle reversal
is an involution on all stopped collisions, so all of them cancel in pairs.
The surviving mass is the exit mass through boundary ports, which equals
the starting mass one. This proves the identity.
For the unsigned bound, close a path ending at b with the counterclockwise
boundary arc from b back to a. This is a simple closed curve, and its total
turning is 2pi, so W(gamma) = pi - (boundary turn from b to a). On a convex
boundary that turn lies in [0, 2pi], hence |W(gamma)| <= pi and
cos(t W(gamma)) >= cos(3 pi/8) = c. Taking real parts in the identity makes
the c-weighted unsigned sum at most one, and division by c bounds the
unsigned sum by 1/c. End of proof.
This identity is the only source of finiteness; every later convergence
claim descends from it. It is the same local cancellation that underlies
the parafermionic observable of Duminil-Copin and Smirnov.
Next come the local weights. A column has N rhombi stacked along horizontal
edges, the second edge of row i making angle u_i/t with the horizontal, and
u_i free. The rhombi are literal for real u_i; for complex values only the
combinatorial content is kept, four ports with a noncrossing pairing and
zero weight for a closed component. With
a(u) = cos(2u - 3 lambda),
L(u) = sin(2 lambda + u) sin(3 lambda + u) = (C + a(u))/2,
the weights are
A(u) = d sin(3 lambda - u) / L(u),
U(u) = d sin(u) / L(u),
B(u) = sin(u) sin(3 lambda - u) / L(u) = (a(u) - c)/(a(u) + C),
E(u) = sin(3 lambda - u) sin(2 lambda - u) / L(u),
F(u) = sin(u - lambda) sin(u) / L(u).
The empty tile has weight 1; a single connection across opposite ports has
weight B; a bottom-left or top-right turn has weight A; the other two
turns have weight U; the two-turns-of-first-kind configuration has weight E
and the second kind weight F. At u = lambda or 2 lambda the rhombus splits
along its short diagonal into two triangles, each visited triangle
contributing rho; at u = 0 or 3 lambda the row is flat, routes pass
deterministically through, and two disjoint routes coexist with weight 1.
These are Nienhuis's integrable O(n) weights at n = 0, the self-avoiding
walk, normalised as in Glazman and Manolescu.
https://doi.org/10.1016/0378-4371(90)90325-M
The diagram algebra has inputs and outputs that are ordered slots, each
vacant or occupied. A diagram is a noncrossing partial pairing of occupied
slots. Composition glues matching slots and annihilates a mismatch or a
closed loop. Tensor product is juxtaposition, and transpose reflects a
diagram, interchanging inputs and outputs. The construction and its
Yang-Baxter operators are algebraic, and the gauge-invariant periodic
partition functions must be distinguished from other gauge-dependent
quantities, as emphasised by Grimm and Pearce.
https://arxiv.org/pdf/hep-th/9303161
The two-slot operator R(v) has inputs (south, west) and outputs (east,
north), with coefficients 1, A(v), B(v), U(v), E(v), F(v) on the empty
diagram, one vertical connection, one diagonal connection, a cap alone, a
cup alone, two vertical connections, and a cap together with a cup. Each
single connection comes in two choices, giving the repeated weights A, A
and B, B and U, U. A left-to-right column uses R(3 lambda - u_i) in row i,
the north output of row i feeding the south input of row i+1; geometric and
operator weights agree because
A(3 lambda - u) = U(u), U(3 lambda - u) = A(u),
B(3 lambda - u) = B(u), E(3 lambda - u) = F(u),
the last also giving F(3 lambda - u) = E(u).
The splitter S_2 maps one slot to two, sending a vacancy to a vacancy plus
rho times a cup and an occupied strand to either output slot with
coefficient rho. The splitter S_3 maps no slot to two, and equals the sum of
the empty diagram and the cup.
Four local identities hold, with R_i acting on slots i, i+1:
(braid) R_2(y) R_1(x+y) R_2(x) = R_1(x) R_2(x+y) R_1(y),
(factorise) R(s lambda) = S_s S_s^T, s = 2, 3,
(split) R_2(x) R_1(x + s lambda)(I (x) S_s)
= (S_s (x) I) Rhat_s(x),
(unitary) R(x) R(-x) = I,
where Rhat_2(x) = R(x + lambda) and Rhat_3(x) is the one-slot identity.
These are identities of rational functions, valid wherever both sides are
regular, including removable singularities. The coefficient checks are
collected at the end.
The dilute-loop structure behind these relations, and its link with
discrete parafermions, was developed by Ikhlef and Cardy, and the
weighted-walk normalisation was treated by Glazman.
https://doi.org/10.1088/1751-8113/42/10/102001
A cut between two columns of N rows has its slots numbered bottom to top.
Let L_N^0 have basis the noncrossing partial pairings of occupied slots,
the rest vacant. Let L_N^1 be the same with one extra unpaired occupied
slot, an exposed defect that no pair may enclose, marking where an exterior
source strand enters. A single arch can meet a cut several times, leaving a
source fragment plus paired fragments on one side, matched on the other.
Before its two auxiliary ends are fixed, a full column is
M_X = R_N(v_N) ... R_1(v_1), v_i = 3 lambda - u_i,
X = (u_1, ..., u_N).
Its ordered inputs are (aux, 1, ..., N) and its ordered outputs are
(1, ..., N, aux): the auxiliary slot slides from the south end to the north
end. Attach a cut state to the west inputs, glue matching occupancies, read
the east state, and give zero to any closed loop. If both auxiliary ends are
vacant, or there is one source, this gives the transfer
T_N^(epsilon) : L_N^epsilon -> L_N^epsilon, with T_N = T_N^(0). A south end
occupied and a north end vacant gives the source insertion
b_N : L_N^0 -> L_N^1, and the opposite choice gives b_N^north. A superscript
minus denotes a right-to-left sweep. Horizontal reflection replaces X by
X* = (3 lambda - u_1, ..., 3 lambda - u_N), so the reflected column is
M_X^- = R_N(u_N) ... R_1(u_1). The empty (all-vacant) coordinate of T_N v
equals that of v, because an occupied input that closed before reaching an
output would form a forbidden loop.
Decay and continuation. If every u_i lies in {0, lambda, 2 lambda,
3 lambda}, then T_N restricted to vectors with zero empty coordinate, and
T_N^(1), both have spectral radius below one. The same holds for reversed
columns and in a complex neighbourhood of each such list, with absolute tile
weights controlling absolute convergence. There is then a unique stationary
vacuum vector P_N with empty coordinate one, rational in the e^(i u_i) and
regular at every such list.
Proof. Start with a homogeneous triangular parallelogram of height H > 0
and width M and a fixed left port. The total weight q_H(M) of paths reaching
the right side tends to zero as M grows. Append a column to the right of the
exit and continue inside it to its bottom port; the continuation uses at
most C_H vertices, depending only on H, so its weight is at least rho^C_H.
For M < L these continuations are distinct paths of the width-L
parallelogram, the bottom endpoint identifying M and the first exit
recovering the original path. The boundary identity on the width-L
parallelogram gives
rho^C_H * sum over M < L of q_H(M) <= 1/c.
As L tends to infinity the sum is bounded and its tail vanishes, so
q_H(M) tends to zero. The same identity bounds paths joining two ports on
one side.
For a list of the four allowed values, collapse the flat rows. The row
vectors lie in {1, e^(i pi/3), e^(2i pi/3), -1}, so outside two end pieces
of horizontal size O(N) the uncollapsed triangles form a homogeneous
parallelogram of height H, the number of nonflat rows, and width M - O(N).
A collapsed deterministic route drifts by at most N columns, and the end
pieces contribute bounded weight.
Every nonzero diagram of a restricted transfer power has a component
joining the left and right cuts: in the vacuum sector otherwise the
nonempty pairing would close a loop, and in the source sector the source
must reach the output. Cut all components at the ends of the homogeneous
core. There are boundedly many endpoint patterns for fixed N, and one
segment spans the core. Ignoring avoidance between segments, which only
increases a positive sum, bounds the total by a product of single-path
partition sums, each nonspanning factor at most 1/c. For H > 0 choose core
width M - kappa_X with 0 <= kappa_X = O(N) fixed. With Q_N the nonempty
vacuum block and max-entry norm,
||Q_N^M||_max + ||(T_N^(1))^M||_max
<= C_N sum over a in E_H of q_{H,a}(M - kappa_X), which tends to 0.
Here E_H is the finite set of core left ports, q_{H,a} its crossing mass
from a, and C_N absorbs the bounded end pieces and endpoint patterns. All
constants are for fixed N, no uniform gap is claimed. If H = 0 no spanning
component exists once M is large. Hence both restricted matrix powers tend
to zero, and as finite matrices their spectral radii are below one.
At the allowed values all tile weights are nonnegative, so absolute tile
weights only increase the radii, and radii move continuously; they stay
below one in a small complex neighbourhood, which also gives absolute
convergence. The reflected argument is identical.
Finally write T_N = [[1, 0], [v, Q]]. The normalised stationary vector is
(1, (I - Q)^(-1) v), regular in the same neighbourhoods and rational in the
spectral variables. This gives uniqueness and continuation; equivalently
P_N is the finite sum of path diagrams in the vacuum half-strip. End of
proof.
Local identities give column identities. Let X' be X with adjacent rows i,
i+1 exchanged and put delta = u_{i+1} - u_i. The braid relation gives
column exchange
M_X (I_aux (x) R_i(delta)) = (R_i(delta) (x) I_aux) M_{X'}.
If u_{i+1} = u_i + s lambda with s = 2 or 3, let X_red replace the two rows
by u_i + lambda when s = 2 or delete them when s = 3, with the splitter S_s
acting on those slots. The splitting identity gives column splitting
M_X (I_aux (x) S_s) = (S_s (x) I_aux) M_{X_red},
its argument being x = v_{i+1}; for s = 2 the effective box has argument
x + lambda = 3 lambda - (u_i + lambda), as required. Transposing and
reversing slot order gives reflected projection
(S_s^T (x) I_aux) M_X^- = M_{X_red}^- (I_aux (x) S_s^T).
All three leave the auxiliary ends untouched, so either vacant ends or
either source condition may be imposed. Writing X as a subscript for its
row list, for epsilon = 0, 1,
T_X^(eps) R_i(delta) = R_i(delta) T_{X'}^(eps),
b_X R_i(delta) = R_i(delta) b_{X'},
T_X^(eps) S_s = S_s T_{X_red}^(eps),
b_X S_s = S_s b_{X_red},
S_s^T T_X^(eps),- = T_{X_red}^(eps),- S_s^T,
S_s^T b_X^- = b_{X_red}^- S_s^T,
and the same with b^north. In a source-insertion identity the right cut map
acts in the vacuum sector and the left one in the one-source sector.
The stationary vector then obeys exchange and reduction:
P_N(..., u_i, u_{i+1}, ...) = R_i(u_{i+1} - u_i) P_N(..., u_{i+1}, u_i, ...),
P_N = S_s P_red when u_{i+1} = u_i + s lambda.
The column equations make the right sides stationary, and their empty
coordinate is one, because a cap turning a nonempty pairing into an empty
output would close a zero-weight loop. Uniqueness in the decay lemma gives
equality, first near the all-zero list (exchange) and near a mixed list with
u_i = 0 (reduction). Intersecting regular neighbourhoods with the
specialisation hypersurface and continuing rationally proves the relations
in general.
Two small facts complete the setup. At u_1 = 3 lambda the first slot is
vacant and may be deleted; at u_1 = 0 the first-slot-vacant part is P_{N-1},
and the occupied part is one column on rows 2, ..., N with a south source
tied to slot 1. These follow from R(0) = I and R(3 lambda) = S_3 S_3^T.
Finally, shifting u_k by pi multiplies each tile weight by the occupancy
signs at its west and east ports, conjugating T_N by the diagonal map D_k
that flips the sign of slot-k-occupied states. Since D_k fixes the empty
coordinate, uniqueness gives P_N(X + pi e_k) = D_k P_N(X).
The stationary vacuum is so far only a rational vector. To read off a
physical observable at coincident parameters u_i = lambda a numerator that
is a polynomial of controlled degree is needed, so that agreement at enough
points forces equality by interpolation. Exchange relations, degree bounds
and reductions at special parameters have this role in earlier dense and
dilute loop calculations.
https://arxiv.org/pdf/math-ph/0410061
https://arxiv.org/pdf/1411.7020
https://arxiv.org/pdf/math-ph/0504032
Write X = (u_1, ..., u_N) and a_i = a(u_i). Define
D(a,b) = a^2 + b^2 + 2 d a b - d^2,
p_N(a_1, ..., a_N) = prod_{i<j} D(a_i,a_j)/(a_i - a_j)
* Pf[ (a_i - a_j)(a_i + a_j + rho) / D(a_i,a_j) ].
For odd N augment the skew matrix by a last column of ones and its negative
transpose, and set p_0 = 1. A Pfaffian is the square root of a
skew-symmetric determinant, Pf[x_ij] = (1/(2^k k!)) sum over pairings of
sign times the product of the entries, with the usual augmentation in the
border case. The reason for this exact object appears when the arch moment
is computed; here its structure is established.
Scalar reduction identities. The function p_N is a symmetric polynomial of
degree at most N-1 in each argument. Its leading coefficient in any
argument is p_{N-1} of the rest, and
p_N(Y, c) = p_{N-1}(Y) prod_{z in Y}(z + C).
Writing f_j = cos(w + j lambda),
p(Y, f_0, f_6) = p(Y)(f_0 + f_6 + rho)
prod_{z in Y}(z - f_{-6})(z - f_{12}),
p(Y, f_{-2}, f_2) = p(Y, f_0)(f_{-2} + f_2 + rho)
prod_{z in Y}(z + f_0).
The subscript of p is its number of arguments, and both formulas also hold
with all shifts negated.
Proof. Degree. After multiplication by all D-factors the Pfaffian is an
alternating polynomial, hence divisible by the Vandermonde product in the
denominator. Counting degrees before and after division bounds the degree
in each variable by N-1.
Leading coefficient. Let one argument tend to infinity. Its kernel entries
tend to -1 and the product factor tends to one, so the leading coefficient
is p_{N-1} of the rest. In odd size the augmenting column and row are first
folded into that argument.
Flat formula. At the flat value c one has D(z,c) = (z - c)(z + C) and the
kernel entry (z - c)(z + c + rho)/D(z,c) equals one. Each term pairing the
flat argument with z acquires a factor (z + C); summing gives the product.
Delete formula. At D(f_0, f_6) = 0 only Pfaffian terms that pair the two
arguments f_0, f_6 survive once denominators are cleared. The identity
D(f_0, z) = (z - f_{-6})(z - f_6) then reproduces the stated product.
Merge formula. Induct on r = |Y|. For r = 0 use p_2(a,b) = a + b + rho.
Assume r >= 1 and single out a spectator z. Both sides have degree at most
r+1 in z, with equal leading coefficient by induction. They agree at
z = -f_0, where deleting this argument with f_{-2} leaves a factor involving
f_2 that vanishes. They agree at z = c by the flat formula and
(C + f_{-2})(C + f_2) = (c + f_0)(C + f_0).
They also agree at each deletion value of z paired with another spectator.
Write that pair as g_0, g_6 with g_j = cos(v + j lambda); then
prod over ell in {+2, -2} of (f_ell - g_{-6})(f_ell - g_{12})
= (f_0 + g_0)(f_0 + g_6)(f_0 - g_{-6})(f_0 - g_{12}),
a product-to-sum identity from shifts by +-2 lambda. There are 2(r-1) such
deletion values besides -f_0 and c. For generic arguments these are
distinct and give at least r+1 roots of a difference of degree at most r,
whose leading coefficient has cancelled. Polynomial continuation removes
the genericity restriction. End of proof.
Wrap the polynomial around the vacuum vector by setting W_N = p_N P_N.
Vacuum degree bound. The vector W_N is Laurent polynomial in
e^(i u_1), ..., e^(i u_N), with every exponent between -(2N-2) and 2N-2.
In the last variable y = u_N and b = a(y), the last-slot-vacant part is a
polynomial in b of degree at most N-1 with leading coefficient W_{N-1}, and
the last-slot-occupied part is sin(y) times a polynomial in b of degree at
most N-2. Set W_0 = 1.
Proof. The case N = 1 is immediate. Induct: build the asserted polynomial,
verify it has no poles and the right degree, check its special-row
relations, and use those relations to prove it is stationary, hence equal to
W_N by uniqueness.
Construction, for fixed earlier rows i < N. Introduce the deletion node and
its cosine values
y_i = u_i + 3 lambda, beta_i = a(y_i),
gamma_i = a(u_i - 3 lambda), sigma_i = a(u_i + 6 lambda).
Row y_i is a deletion partner of row i, separated by 3 lambda in the
a-variable. Let R_i^move move that row from just above row i to the last
position through successive exchange boxes with arguments y_i - u_j,
i < j < N. By the vacuum exchange and reduction identities and the delete
formula, the value the candidate must take at y = y_i is
Y_i = (a_i + beta_i + rho)
prod over j < N, j != i of (a_j - gamma_i)(a_j - sigma_i)
times R_i^move S_3 W_{N-2}(X without {u_i, u_N}).
These node data agree with p_N P_N on the node hypersurface: check the
identity near u_i = 0 with the other rows mixed, where all stationary
vectors are regular. That mixed point gives an open neighbourhood inside
the hypersurface, hence a rational identity in the remaining parameters,
not equality at a single point.
For distinct nodes put the Lagrange cardinal polynomials
ell_i(b) = prod over j < N, j != i of (b - beta_j)/(beta_i - beta_j),
and define the candidate by the two interpolation formulas
W_v(b) = W_{N-1} prod over i < N of (b - beta_i)
+ sum over i < N of (Y_i)_v ell_i(b),
W_o(y) = sin(y) sum over i < N of (Y_i)_o / sin(y_i) ell_i(b).
The vacant part has degree at most N-1 and leading coefficient W_{N-1}; the
occupied part divided by sin(y) has degree at most N-2; both take the
prescribed node values. Scalar interpolation shows the empty coordinate
of W is p_N.
There are no poles in earlier variables. Each moving box carries two sine
factors in its denominator, and they divide the factors
(a_j - gamma_i)(a_j - sigma_i) in the node data. The possible divisor
sin(y_i) in the occupied data also cancels. Explicitly, with x = u_i and
v = u_j,
(a(v) - a(x - 3 lambda))(a(v) - a(x + 6 lambda)) / L(x + 3 lambda - v)
= -4 sin(x + v - 6 lambda) sin(x + v + 3 lambda),
a(x) + a(x + 3 lambda) + rho
= 4 c sin(x + 3 lambda) sin(3 lambda - x).
The remaining possible divisors come from colliding nodes, since
beta_i - beta_j = -2 sin(u_i + u_j + 3 lambda) sin(u_i - u_j),
so beta_i = beta_j splits into u_i = u_j mod pi and
u_i + u_j = -3 lambda mod pi. On the first component the data, including the
divided occupied data, agree with the true vector and its occupancy signs,
verified near the regular mixture u_i = u_j = 0, y_i = y_j = 3 lambda. On
the second component both data vanish through their sigma_i, sigma_j
factors, and at a generic point no moving-box denominator vanishes there.
These checks at generic points of every irreducible divisor suffice, since
all denominators are products of the indicated simple Laurent factors. The
candidate is therefore Laurent polynomial in the earlier variables, with no
new denominators at multiple intersections.
For the degree, send an earlier u_k to either Laurent extreme. The moving
boxes stay bounded there, directly from the tile weights. Data with i != k
grow at most like a_k^(N-1): they carry two scalar factors and the inductive
vector W_{N-2}, while ell_i(b) stays bounded. Data with i = k grow at most
like a_k^(1+2(N-2)), while their Lagrange cardinal contributes
O(a_k^(-(N-2))). The leading term W_{N-1} prod(b - beta_i) obeys the same
bound. With poles removed, growth at the two Laurent extremes bounds every
earlier exponent between -(2N-2) and 2N-2. A shift u_k -> u_k + pi
preserves the occupancy sign of the interpolant; when it hits a node, the
last-slot sign cancels the sign of the dividing sine.
Earlier-row braid covariance follows by last-variable interpolation: the
exchange box is independent of the last variable, the leading coefficients
agree by induction, and each node datum agrees by the rational vacuum
exchange identity, so uniqueness of the interpolant gives covariance.
With the degree bounds and node values established, stationarity needs more
zeros than last-variable interpolation provides. Two row specialisations
supply them.
First, if adjacent earlier rows satisfy u_{r+1} = u_r + s lambda with
s = 2 or 3, the interpolant equals the smaller split vector times the scalar
quotient from the delete or merge formula. That quotient has degree one in b
for merge or two for delete, with monic last-variable factor, so the degree
forms and leading coefficient agree by induction. At a node indexed by
i != r+1 compare with the true vector near a regular simultaneous
specialisation: u_r = 0, u_i = 0 when distinct, y_i = 3 lambda, others
mixed. At i = r+1 both sides vanish. On the split side the vanishing factor
is b - sigma_r for delete or b + a(u_r + lambda) for merge; on the node-data
side it is a_r - gamma_i or a_r - sigma_i. No moving box crosses row r.
Second, at u_1 = 0 or 3 lambda use the first-slot formulas, replacing the
smaller vacuum vector by W_{N-1} and multiplying by prod_{j=2}^N (a_j + C).
The only degree check needing care is the south-source column in the
occupied first-slot part. With north vacant the last box contributes 1 or
A(y) to a vacant output and U(y) or B(y) to an occupied output; multiplying
by b + C clears the denominator and gives the required last-variable forms.
Only the contribution 1 reaches leading vacant degree, leaving the formula
with the last row omitted. At nodes the expressions agree at regular mixed
specialisations, taking u_i = 0 for i > 1. The exception u_1 = 3 lambda,
i = 1, y_i = 6 lambda is handled by the vanishing factors a(y_i) + C and
a_1 + beta_1 + rho.
On each specialisation the nodes are generically distinct with nonzero
dividing sines, and the remaining cases follow by continuation.
Stationarity. Form the polynomial residual
V = ( prod over j = 1 to N of L(u_j) ) (T_N - I) W.
At each last-variable node it vanishes, since the node data came from the
genuine stationary vector. To control its degree put n = N-1,
q = W_o/sin(y), Lambda_n = prod over j < N of L(u_j). Besides the
north-source map b_n^north use C_n^north : L_n^1 -> L_n^0, obtained from a
column with north occupied and south vacant by tying its north end to the
exterior defect of the input state; an output that closes a loop
contributes zero. These maps and the transfers on the first n rows do not
depend on y.
The last box contributes 1 or A(y) to a vacant output and U(y) or B(y) to
an occupied one. With L(y) = (b + C)/2 and
sin(y) sin(3 lambda - y) = (b - c)/2 this gives the exact blocks
V_v = (Lambda_n/2) ( (b + C)(T_n - I) W_v
+ d (b - c) C_n^north q ),
V_o/sin(y) = (Lambda_n/2) ( (b - c) T_n^(1) q - (b + C) q
+ 2 d b_n^north W_v ).
The second block has degree at most N-1. In the first, the only possible
degree-N term has coefficient
[b^N] V_v = (Lambda_n/2)(T_n - I) W_n = 0
by inductive stationarity, so V_v also has degree at most N-1. Every
coordinate of V is therefore divisible by prod over i < N of (b - beta_i).
In each earlier variable u_k its Laurent exponents are bounded by 2N. The
two specialisations make V vanish when an earlier row is 0 or 3 lambda, or
when u_j = u_i + s lambda with i < j < N and s = 2 or 3; to pass from
adjacent to arbitrary pairs move the two rows together without interchanging
them, using earlier-row covariance and the column identity, generic
invertibility following from unitarity.
Fix k < N. These zeros force divisibility by
(b - beta_k) sin(u_k) sin(u_k - 3 lambda)
prod over j < k, s in {2,3} of sin(u_k - u_j - s lambda)
prod over k < j < N, s in {2,3} of sin(u_j - u_k - s lambda).
For generic other variables this divisor has Laurent extremes +-2N: the
factor b - beta_k contributes two in each direction, the two flat sine
factors contribute two, and the 2(N-2) pair factors the remaining 2N-4.
Its parity is even under u_k -> u_k + pi. A Laurent polynomial with the
same exponent bound divided by this divisor is independent of e^(i u_k).
But if slot k is occupied, the coordinate of V has odd parity, so it
vanishes. Every nonempty vacuum pairing has an occupied earlier slot, as it
has at least two occupied slots, and the empty residual is identically zero.
Hence V = 0 and W is stationary. Its empty coordinate is p_N, so uniqueness
at generic regular parameters gives W = p_N P_N. End of proof.
The polynomial vacuum now returns to a positive path observable. Cutting a
rightward arch at every column between its endpoints counts it exactly as
many times as its horizontal displacement, which turns the moment into a
pairing of two source states and then into one coefficient of the ratio
p_{N+1}/p_N.
For a repeated rhombic column X near a physical list, let K_X(k) be the
bottom-to-bottom partition sum with displacement k, with exactly its two
endpoints occupied on the exterior boundary and one connected path, and set
m(X) = sum over k >= 1 of k K_X(k). A cut can meet one arch several times,
each half holding the source fragment plus paired fragments. With the
south-source column b_N put
H_N = (I - T_N^(1))^(-1) b_N P_N,
and H_N^- in the reflected strip. Here P_N supplies the paired fragments
before the source column and the resolvent sums the columns between that
source and the cut. Let Glue be the bilinear pairing of one-source states
that equals one when the occupancies agree and gluing joins the two sources
into one component with no loops, and zero otherwise. Then
m(X) = Glue(H_N^-, H_N).
Expanding the two resolvents puts one source on each side of the cut, and an
endpoint separation k offers exactly k cuts between them. The decay lemma
justifies absolute convergence of the source and vacuum sums in a
neighbourhood of every mixed list, so m is rational in the spectral
variables and regular at all such lists.
Related finite-strip current calculations use bilinear link-state pairings,
fusion reductions and interpolation, though those O(1) current formulas
assume a cancellation of a normalising factor or an extra symmetry.
https://arxiv.org/pdf/1004.4037
https://arxiv.org/pdf/1510.02721
Arch formula. Let eta = 2d - 1 = sqrt 2 - 1, so that c = eta C and
C = eta(2C + c), and let
g_X(z) = p_{N+1}(a_1, ..., a_N, z) / p_N(a_1, ..., a_N).
For generic X, g_X is monic of degree N, and
m(X) = sum over i = 1 to N of a_i - eta [z^(N-1)] g_X(z),
the right side understood by continuation at removable singularities.
Proof. The moment is identified from its row reductions together with a
separate degree bound of 2N for p_N^2 m in each a_i. The source-sector limit
supplies that bound, and the scalar reductions then fix the coefficient.
Symmetries and reductions. Moving an exchange box through both halves of
the gluing identity, using the column intertwiners, proves symmetry in the
rows. At an ascending merge or deletion specialisation the same identities
for the vacuum and south-source columns give P_N = S_s P_red and
H_N = S_s H_red, while the reflected column identity gives
S_s^T P_N^- = P_red^- and S_s^T H_N^- = H_red^-. The first projected vector
is stationary with empty coordinate one. The second satisfies
(I - T_red^(1),-) S_s^T H_N^- = b_red^- S_s^T P_N^- = b_red^- P_red^-,
so by uniqueness near a mixed list and rational continuation it equals
H_red^-. Transposing the splitter inside the gluing pairing reduces m to the
smaller strip, the two halves reducing by insertion and by projection; no
identity S_s^T S_s = I is needed. Occupancy signs cancel under a pi shift of
a parameter.
There is also a reflection u_j <-> 3 lambda - u_j. By exchange put that row
on top, where every port is vacant, so each visit to the row is an interval
turning up then down. An interval with displacement ell >= 1 has weight
A U B^(ell-1), invariant under this reflection, and the intervals are
ordered and disjoint. The argument holds for absolutely convergent sums and
hence as a rational identity. If a_j = c then A U = 0 and these excursions
vanish, so the flat row may be omitted.
A polynomial bound from the source sector. Use a north source. Add a last
row y to the vacuum strip and identify its occupied last-slot sector with
one-source states on the earlier rows. Last-box stationarity reads
(P_{N+1})_o = B(y) T_N^(1)(P_{N+1})_o + U(y) b_N^north (P_{N+1})_v.
As b = a(y) grows, the two lemmas give (P_{N+1})_v -> P_N and B(y) -> 1, so
(P_{N+1})_o/U(y) tends to the integrated north-source vector. Since
U(y) = 2d sin(y)/(b + C), its numerator after multiplication by p_N is
(1/(2d)) [b^(N-1)] (W_{N+1})_o / sin(y).
The degree bound holds its Laurent exponents in each earlier variable by
2N. Reflection top to bottom reverses row order and sends u_i to
3 lambda - u_i, preserving p_N and those bounds, so both p_N H_N and p_N H_N^-
obey the same bound. By the gluing identity p_N^2 m has exponent bound 4N.
Its separate reflection and pi-periodicity make it a polynomial in
a_i = cos(2u_i - 3 lambda) of degree at most 2N in each variable: a
pi-periodic Laurent polynomial is a Laurent polynomial in z = e^(2i u_i),
and invariance under z -> e^(6i lambda)/z makes it a polynomial in the
scaled sum a_i. Row exchange makes it symmetric.
Interpolation. Let the right side of the arch formula be m~. The scalar
identities give exact quotient rules
g_merge(z) = (z + f_0) g_red(z),
g_delete(z) = (z - f_{-6})(z - f_{12}) g_red(z),
g_flat(z) = (z + C) g_red(z).
The elementary identities f_{-2} + f_2 - f_0 = eta f_0,
f_0 + f_6 = -eta(f_{-6} + f_{12}), and c = eta C show that m~ has the same
three reductions as m. Both p_N^2 m and p_N^2 m~ have degree at most 2N in
each a_i. Induct on N. In one distinguished variable, every other argument
a_j = cos(theta_j) supplies the two merge values cos(theta_j +- 4 lambda) and
the two deletion values cos(theta_j +- 6 lambda), plus the flat value c.
These are 4N - 3 distinct roots for generic other arguments, more than 2N for
every N >= 2, so the polynomial difference vanishes. The initial cases are
m(empty) = 0 and
m(u) = A(u)U(u)/(1 - B(u))^2 = (a(u) - c)/(1 + d)
= a(u) - eta(a(u) + rho).
In the size-zero quotient convention use g_0 = 1 and [z^(-1)] g_0 = 0. This
completes the induction. End of proof.
At X = (lambda, ..., lambda) each rhombic row splits into the two triangles
of one band with weight rho per visited vertex, so a_i = C and m(X) = m_N.
In particular
m_N = N C - eta lim_{a_i -> C} [z^(N-1)] g_X(z),
the limit being removable because m is regular at the physical list. Later
the whole monic polynomial g_X is shown to have a unique limit, and the
coefficient is evaluated through a positive integral. A strip-geometry
estimate first compares successive arch moments with the bridge mass.
Moment increments and bridge mass. For homogeneous honeycomb strips,
c A_N + B_N = 1, B_N is nonincreasing, and m_{N+1} - m_N is comparable to
B_N, uniformly in N >= 1.
Proof. Apply the boundary identity to parallelogram truncations of the
strip. The left and right boundary terms tend to zero by the decay lemma:
after the source column propagate in the one-source sector to the distant
cut, where the exit is the only occupied boundary port, and the remaining
vacuum states are bounded. The other sums increase to their infinite-strip
limits. Bottom exits have winding +-pi and top exits winding zero, so the
real part is
c A_N + B_N = 1.
Increasing the strip height increases the arch sum by inclusion, hence
decreases B_N.
Now compare heights. Add one triangular row above the right half of S_N, in
columns 1, 2, ..., forming a stepped strip with a single side port p at the
left end of the new row, and denote its bottom arch kernel by Kbar(i,j).
From a bottom source i the imaginary part of the boundary identity reads
sin(3 lambda) ( sum over j < i of Kbar(i,j)
- sum over j > i of Kbar(i,j) )
+ sin(lambda) Z(i,p) = 0.
Horizontal top exits have winding zero, and closing along the leftward
boundary gives winding pi/3 at p. Lateral truncation errors vanish as
before, dominated by a strip of height N+1.
Sum this over -L <= i <= L. Reversal pairs and cancels all internal arches,
leaving uncancelled terms crossing the two ends of the interval. After
translating each end to the origin the stepped domains converge locally to
strips of heights N and N+1. All crossing sums are dominated by the first
moment in the strip of height N+1, finite by decay, and paths leaving an
arbitrarily large lateral window are controlled for each fixed pair of
ports. Dominated convergence gives the exact identity
m_{N+1} - m_N = (sin(lambda)/sin(3 lambda)) sum over i of Z(p,i).
Lower bound. Turning down immediately from p and following an arbitrary
bridge in the base strip gives sum over i of Z(p,i) >= rho B_N. The
top-source bridge mass in the base strip equals the bottom-source mass by
translation and reversal.
Upper bound. Order the path's visits to the interface between base strip and
new row from left to right. In the new row, whose top is vacant, disjoint
intervals pair p with the first visit and then pair subsequent visits
consecutively. The base portions form a noncrossing pairing with one unpaired
visit connected to the bottom, a defect that cannot be enclosed. Since the
union is a single path, the first base visit pairs with its next neighbour,
because pairing it farther would leave a closed component inside; removing
these two visits and repeating shows base visits also pair consecutively,
with the last visit the defect. Interface visits thus occur in increasing
horizontal order. At u = lambda the initial new-row segment has weight at
most A/(1 - B). A one-sided base return arch has mass at most 1/(2c), by the
boundary identity from the top and equality of the two directions, and a
one-sided return through the new row has mass A U/(1 - B) = 2cB. The final
base segment has mass B_N. Ignoring avoidance between pieces only increases
the sum, since base weights are products over vertices. Therefore
sum over i of Z(p,i)
<= (A/(1 - B)) sum over r >= 0 of ((1/(2c)) 2cB)^r B_N
= (A/(1 - B)^2) B_N.
Here B = B(lambda) = rho^2 < 1, so the constants are independent of N.
Combine with the exact identity. End of proof.
The arch formula is exact, but the Pfaffian quotient does not directly show
the size of m_N. The coalescing-parameter problem converts into a positive
integral, which proves the limiting quotient exists without assuming the
denominator stays nonzero and opens the moment to comparison estimates. For
generic X write b = cos(w) and
Q_X(b) = (b - C) g_X(b)
prod over i of (b - a_i)(cos(w - 2 lambda) - a_i)(cos(w + 2 lambda) - a_i),
Phi_X(w) = sin(w) Q_X(cos(w)).
The paired shifted factors have product b^2 - 2 d a_i b + a_i^2 - d^2, so
Q_X is a polynomial of degree 4N + 1.
Missing harmonics. Every sine harmonic of Phi_X whose frequency is divisible
by four has coefficient zero. Equivalently, with h = b^2(1 - b^2), which is
(1 - cos 4w)/8, there are polynomials with
Q_X(b) = P(h) + b J(h) + b^2 E(h),
deg P, deg J <= N, deg E <= N - 1.
Proof. Fix i and write f_j = cos(v + j lambda) with f_0 = a_i. The product
in Q_X gives Phi_X(v +- 2 lambda) = 0, and the delete quotient gives
Phi_X(v + 6 lambda) + Phi_X(v - 6 lambda) = 0. Cancelling the contributions
of the other a_j, which on both sides involve exactly f_6, f_{-6}, f_4,
f_{-4}, -f_0, leaves
R_+ = sin(v + 6 lambda)(f_6 - C)(f_0 + f_6 + rho)(f_6 - f_0)(f_4 - f_0),
R_- = sin(v - 6 lambda)(f_{-6} - C)(f_0 + f_{-6} + rho)(f_{-6} - f_0)
(f_{-4} - f_0).
Product-to-sum and rho = 2cd give
R_+ = cos(2v)(f_6 - C) sin(v + 3 lambda)(cos(v + 3 lambda) + d),
R_- = cos(2v)(f_{-6} - C) sin(v - 3 lambda)(cos(v - 3 lambda) + d).
After removing cos(2v) each is a trigonometric polynomial of degree three
with the same six zeros v = +-3 lambda, +-5 lambda, +-7 lambda modulo 2pi
and opposite values at zero, hence they are negatives. This proves the
cancellation as a polynomial identity, including where cancelled factors
vanish.
Add now the four shifts of Phi_X by +-2 lambda and +-6 lambda. On frequency
k the multiplier is 4 cos(4k lambda) cos(2k lambda), nonzero exactly when 4
divides k, and the maximum frequency is 4N + 2. Dividing the shifted sum by
sin(4w) gives a polynomial in cos(4w) of degree at most N - 1, and the
preceding identities give N distinct roots for generic a_i, so it is zero.
That is the harmonic assertion.
Every polynomial in b has a unique expression P_0(h) + b P_1(h) + b^2 P_2(h)
+ b^3 P_3(h), by repeatedly using b^4 = b^2 - h. The first three terms,
times sin(w), carry no frequency divisible by four. For the last use
sin(w) b^3 = (1/4) sin(2w) + (1/8) sin(4w),
whose projection onto multiples of four is (1/8) sin(4w) P_3((1 - cos 4w)/8),
vanishing only if P_3 = 0. The degree bounds follow from deg Q_X = 4N + 1.
End of proof.
At the homogeneous parameters a_i = C the paired factors become (b - C)(b - c),
so the limiting monic polynomial is characterised by
Q(b) = (b - C)^(2N+1) (b - c)^N g(b)
= P(h) + b J(h) + b^2 E(h), deg g = N.
Nonsingularity and integral representation. The monic problem above has a
unique solution, and it is the limit of g_X as all a_i tend to C. Put
omega(x) = (1/2) sqrt( sqrt((1 + x)/(2x)) - 1 ), 0 < x < 1,
and let (y_1, ..., y_N) have density proportional to
prod over i of omega(y_i) prod over i < j of (y_j - y_i)^2/(y_i + y_j),
0 < y_1 < ... < y_N < 1.
Then
m_N = (eta/pi) int_0^1 ( 1 - E prod over i of (y_i - x)/(y_i + x) )
omega(x)/x dx,
the expectation taken over the positive law.
Proof. Well-definedness. Since omega(x) = O(x^(-1/4)) at zero and each pair
factor is at most y_j <= 1, the normalising integral is finite and positive.
Cauchy moment equations. The two prescribed zeros satisfy h(C) = h(c) = 1/8.
Set z = (8h - 1)^(-1) to send them to infinity, with inverse relation
b^2(1 - b^2) = (1 + 1/z)/8. Use the two branches near infinity tending to C
and c; their differences from these limits are of order z^(-1), because
h'(C) and h'(c) are nonzero. Multiply each polynomial in the homogeneous
relation by z^N after substituting h = (1 + 1/z)/8, calling the results Pt,
Jt, Et, with Et(0) = 0 since deg E <= N - 1. The branch tending to C and the
sum of the two branches are
f(z) = sqrt( (1 + sqrt((1 - 1/z)/2))/2 ),
l(z) = sqrt( 1 + sqrt((1 + 1/z)/2) ).
f continues off [0, 1] and l off [-1, 0]. On the C branch the zero of order
2N + 1 becomes O(z^(-2N-1)); on the c branch the zero of order N becomes
O(z^(-N)). After multiplication by z^N,
F(z) = Pt(z) + f(z) Jt(z) + f(z)^2 Et(z) = O(z^(-N-1)),
D(z) = Jt(z) + l(z) Et(z) = O(1),
the second obtained by evaluating on the c branch, which is O(1) after
multiplication by z^N, and subtracting the C-branch expression.
With subscripts + and - for boundary values from the two half-planes,
f_+(x) - f_-(x) = 2i omega(x),
l_+(-x) - l_-(-x) = -2 sqrt(2) i omega(x),
f_+(x) + f_-(x) = l(x).
These are bounded at the outer endpoints. At zero F = O(|z|^(-1/4)) and
D(z) tends to D_0 = Jt(0). Cauchy's formula, with small endpoint circles
shrinking to zero, gives
F(z) = (1/pi) int_0^1 D(x) omega(x)/(x - z) dx,
int_0^1 x^j D(x) omega(x) dx = 0, 0 <= j < N,
D(z) = D_0 - (sqrt(2)/pi) z int_0^1 (Et(-y)/y)/(z + y) omega(y) dy.
The moment equations are the first N coefficients of F at infinity; the
third display subtracts the bounded-at-infinity representation at zero, its
integrand integrable there because Et(-y)/y is a polynomial. Near zero the
increment in the third line, split at y = x, is bounded by
C( int_0^x y^(-1/4) dy + x int_x^1 y^(-5/4) dy ) = O(x^(3/4)).
Nonsingularity. The polynomial Et(-y)/y has degree at most N - 1 and the
moment equations determine it uniquely from D_0. The corresponding matrix,
up to nonzero scalars,
M_jk = int_0^1 int_0^1 x^(j+1) y^k/(x + y) omega(x) omega(y) dx dy,
0 <= j, k < N,
has strictly positive determinant. This is Cauchy-bimoment positivity, the
finite determinant version due to Bertola, Gekhtman and Szmigielski; expanding
and antisymmetrising in x and y turns the integrand into the product of two
positive Vandermonde determinants, prod x_i, positive weights, and
det[1/(x_i + y_j)] = prod_{i<j}(x_j - x_i)(y_j - y_i) / prod_{i,j}(x_i + y_j),
which is positive by clearing denominators and checking the constant one.
https://arxiv.org/pdf/0904.2602
A solution of the homogeneous problem for g, of degree at most N - 1, has
[b^(4N+1)] Q = 0, but [b^(4N+1)] Q = (-8)^N D_0, so D_0 = 0. Positivity of M
forces Et = 0, then Jt = 0, then Pt = 0, hence g = 0. For the monic problem
write g(b) = b^N + sum_{r<N} gamma_r b^r; the excluded b^3 P_3(h) part has N
coefficients, whose vanishing gives an N by N system that the injectivity
makes invertible. Applying the same coefficient conditions to Q_X with a
monic unknown polynomial gives a system depending polynomially on the a_i
and converging to the invertible homogeneous one, so it is invertible for all
a_i close to C, and the generic g_X converges to g. The moment itself is
regular at the homogeneous point by the decay lemma.
The moment coefficient. For the monic solution D_0 = (-8)^(-N); put
G(z) = D(z)/D_0. The leading-coefficient computation gives
[b^(4N)] Q = (-8)^N Pt(0), and comparing with the product in the homogeneous
relation gives
[b^(N-1)] g(b) = Pt(0)/D_0 + (2N + 1) C + N c.
Substituting into the physical arch formula, using C = eta(2C + c), yields
m_N = -eta ( C + Pt(0)/D_0 )
= (eta/pi) int_0^1 (1 - G(x)) omega(x)/x dx.
For the second equality use f(z) = C + (1/pi) int_0^1 omega(x)/(x - z) dx,
subtract D_0 f from the Cauchy representation of F, and let z tend to 0 from
below; the third display gives D(x) - D_0 = O(x^(3/4)), so the subtracted
integrand is O(x^(-1/2)) at zero, integrable. On the left the terms with
f(z)(Jt(z) - D_0) and f(z)^2 Et(z) tend to zero.
It remains to identify G with the expectation. Let W_n be the product in the
positive law with n coordinates, on (0,1)^n without ordering, normalised.
Partial fractions give
prod over i of (y_i - x)/(y_i + x) = 1 - x sum over i of alpha_i/(x + y_i),
alpha_i = 2 prod over j != i of (y_j + y_i)/(y_j - y_i).
After multiplying by the density, the factors in alpha_i cancel one
Vandermonde and all matching denominators, and each cancelled term is
integrable: remaining polynomial factors and pair kernels are bounded on
(0,1)^N, as is x/(x + y_i), and the one-variable weights are integrable.
Integrating the other variables leaves 1 - x int_0^1 P(y) omega(y)/(x + y) dy
with deg P <= N - 1, whose moments against x^k omega(x) vanish for k < N:
adjoin x as an (N+1)-st variable and use the density identity
x^k omega(x) W_N(y) prod over i of (y_i - x)/(y_i + x)
= x^k W_{N+1}(x, y_1, ..., y_N) / prod over i of (y_i - x).
The left side is absolutely integrable, the product having modulus at most
one, and symmetrising the right side over all N+1 variables gives zero by the
elementary interpolation identity
sum over r of z_r^k / prod over j != r of (z_j - z_r) = 0, 0 <= k < N.
The expectation therefore has the same polynomial Cauchy representation and
vanishing moments as G, and uniqueness identifies it with G. Substitute into
the moment-D formula. End of proof.
The positive integral reduces the growth of the arch moment to the
behaviour of the coordinates nearest zero. Those are bounded by comparing
with an explicitly normalised Bures-Laguerre density; the comparison is
between positive measures, so it also controls sign changes in the product.
The integral formula gives at once
E y_1^(-1/4) ~< m_N ~< E ( sum over i of y_i^(-1) )^(1/4).
Put R(x) = prod over i of (y_i - x)/(y_i + x). Each factor lies in [-1, 1],
so 1 - R(x) >= 0, and telescoping the product gives
1 - R(x) <= 2 min{ 1, x sum over i of y_i^(-1) }.
For y_1/3 < x < y_1/2 all factors are nonnegative and the first is at most
1/2, so 1 - R(x) >= 1/2. Also omega(x) ~< x^(-1/4) on (0,1) and
omega(x) ~ x^(-1/4) on (0, 1/2). Integrating the lower bound on that
interval and splitting the upper integral at x = (sum y_i^(-1))^(-1) proves
the comparison.
The reference inverse-moment estimates below contribute a factor N^(1/2),
and scaling the coordinates by a factor of order N^(-1) contributes a
further N^(1/4), giving N^(3/4) for the arch moment.
The reference law is the generalised Bures-Laguerre family, in ordered
coordinates with parameter a > 0, of density
(1/Z_n(a)) prod over i of r_i^(a-1) e^(-r_i)
prod over i < j of (r_j - r_i)^2/(r_i + r_j),
0 < r_1 < ... < r_n < infinity.
Expectations and probabilities for this law carry subscripts n, a. The
one-variable weight exponent is a - 1, so this a exceeds the parameter of
Forrester and Kieburg by one; the ordered normaliser is the unordered
integral divided by n!.
https://arxiv.org/pdf/1410.6883
Association and monotone comparison. Let I be an interval in (0, infinity),
n >= 1, and w_1, ..., w_n positive smooth on I. If
prod over i of w_i(x_i) prod over i < j of K(x_i, x_j),
K(x, y) = (y - x)^2/(x + y), x_1 < ... < x_n, x_i in I,
has finite positive normalising integral, then its probability law is
positively associated: increasing integrable functions have nonnegative
covariance whenever defined. The same holds after truncation to a shorter
common interval. Consequently an increasing likelihood ratio shifts the
coordinates up in stochastic order and a decreasing one shifts them down.
Proof. The mixed derivative of each pair is
d^2/(dx dy) log K(x, y) = 2/(y - x)^2 + 1/(x + y)^2 > 0.
Proceed by induction on the number of coordinates, first for bounded
increasing functions. Conditional on the largest coordinate t, the rest have
a law of the same form on the shorter interval with modified one-variable
factors, so they are associated by induction. For s < t the ratio of the
conditional density at s to that at t, on the former support, is
coordinatewise decreasing by the mixed-derivative inequality, and extended
by zero where a coordinate exceeds s it stays decreasing. Association at t
makes a decreasing likelihood tilt decrease every increasing expectation;
an unbounded ratio is truncated and handled by monotone convergence. Hence
the conditional expectation of every increasing function of all coordinates
increases with t.
The law of total covariance then splits into two nonnegative terms:
conditional covariance by induction, and covariance of the two conditional
expectations, nonnegative by
Cov(F(T), G(T)) = (1/2) E[ (F(T) - F(T'))(G(T) - G(T')) ] >= 0,
T' an independent copy of T. Truncation and integrable approximation extend
the result to the stated class. Finally, for an increasing likelihood ratio
L, E[F L]/E L >= E F for increasing F, with the reversed inequality for
decreasing L, which is the stochastic comparison. End of proof.
This is the continuous MTP_2 principle of Sarkar, stated for log-supermodular
densities by Bogso and by Fallat, Lauritzen, Sadeghi, Uhler, Wermuth and
Zwiernik; the conditioning proof above handles the ordered pair kernel needed
here.
https://doi.org/10.1051/ps/2013049
https://doi.org/10.1214/16-AOS1478
The upper comparison uses a = 1/2, the lower comparison a = 1, and the
minimum at a = 1 needs an inverse-moment bound at a = 2. All three come from
one normalising integral.
Laguerre normalization and coordinate bounds. For fixed a > 0 and every
n >= 1,
Z_n(a) = prod over i < n of Gamma(a + i)
prod over 0 <= i < j < n of (j - i)/(2a + i + j).
There are constants K_a, C_a depending only on a with
Prob_{n,a}(r_n <= K_a n) bounded below independently of n, and, with
D_k = a k + k(k-1)/2,
Prob_{n,a}(r_k < h) <= min{ 1, ( C_a h n^2/k^3 )^(D_k) }, h > 0.
In particular
E_{n,1/2} ( sum over i of r_i^(-1) )^(1/4) ~< n^(1/2),
E_{n,2} sum over i of r_i^(-1) ~< n^2,
E_{n,1} r_1^(-1/4) ~> n^(1/2),
with absolute constants in the last three bounds.
Proof. Normalization. Write the pair product as a Vandermonde determinant
times the Schur Pfaffian with entries (s - r)/(s + r); the identity
Pf[ (x_j - x_i)/(x_j + x_i) ] = prod over i < j of (x_j - x_i)/(x_j + x_i)
is classical. For even size, clearing denominators leaves an alternating
polynomial of Vandermonde degree, so only its constant needs checking, and
pairing two opposite arguments gives that constant recursively as one; for
odd size add an argument tending to infinity, whose augmenting column
consists of ones.
https://arxiv.org/pdf/math/0411280
Expand the determinant and the Pfaffian and integrate over the unordered
region, divided by n!. This is de Bruijn's determinant-Pfaffian integration
formula, with the general antisymmetric-kernel form including its odd-size
border; absolute integrability follows from |(s - r)/(s + r)| <= 1 and finite
gamma moments for a > 0.
https://pure.tue.nl/ws/files/1920642/597510.pdf
https://arxiv.org/pdf/math/9905083
The double moment with indices i, j is
int int r^(a+i-1) s^(a+j-1) e^(-r-s) (s - r)/(s + r) dr ds
= Gamma(a+i) Gamma(a+j) (j - i)/(2a + i + j),
obtained from r + s and r/(r + s), the latter beta integral having mean
(a + i)/(2a + i + j). Factoring out the gamma terms and using the Schur
identity again at the arguments a + i proves the normaliser formula.
Coordinate tails. For the maximum, condition on its value s; every factor
involving a smaller coordinate is at most s, since K(r, s) <= s for
0 < r < s. Hence
Prob_{n,a}(r_n > K n)
<= (Z_{n-1}(a)/Z_n(a)) int_{Kn}^{inf} s^(a+n-2) e^(-s) ds,
and the explicit normaliser gives
Z_{n-1}(a)/Z_n(a)
= (1/Gamma(a+n-1)) prod over j < n of (2a + 2n - 2 - j)/j
<= e^a (2e)^(n-1)/Gamma(a+n-1),
by replacing every numerator with 2(a + n - 1), using
(n-1)! >= ((n-1)/e)^(n-1), then (1 + a/(n-1))^(n-1) <= e^a. The
exponential-moment bound for a gamma variable gives
(1/Gamma(a+n-1)) int_{Kn}^{inf} s^(a+n-2) e^(-s) ds
<= e^(-Kn/2) 2^(a+n-1),
so a large K makes the tail exponentially small in n.
Small coordinates. Integrating the first k coordinates below h, their
mutual pair factors are at most h and a factor between a small coordinate
and a remaining one is at most the latter, so discarding the order
constraints between the two groups gives
Prob_{n,a}(r_k < h)
<= h^(D_k)/(a^k k!) Z_{n-k}(a+k)/Z_n(a), Z_0 = 1.
The normaliser formula gives
Z_{n-k}(a+k)/Z_n(a)
= prod over i < k of Gamma(a+i)^(-1)
prod over i < k, i < j < n of (1 + (2a + 2i)/(j - i)).
For gaps j - i <= k, 2a + 2i <= C_a k and
sum_{j<=k} log(1 + C_a k/j) = O_a(k), so the short gaps contribute
O_a(k^2) = O_a(D_k). For larger gaps, log(1 + x) <= x gives at most
2 D_k log(n/k). And
prod over i < k of Gamma(a+i) >= k^(D_k) e^(-O_a(D_k)),
since integration on [q, q+1] gives log Gamma(q) >= (q-1) log q - q - O_a(1)
for q = a + i, and summing with sum (a+i) log((a+i)/k) >= -k^2/e yields
D_k log k - O_a(D_k). Absorbing a^(-k)/k! into C_a^(D_k) proves the
small-coordinate bound.
For 0 < p < D_k, integrating the tail gives
E_{n,a} r_k^(-p) <= (D_k/(D_k - p)) ( C_a n^2/k^3 )^p.
At a = 1/2 with (k, p) = (1, 1/4), and then p = 1 for k >= 2, this gives
E r_1^(-1/4) ~< n^(1/2) and E sum_{k>=2} r_k^(-1) ~< n^2; subadditivity of
the fourth root and Jensen give the first display. At a = 2, p = 1 for every
k gives the second.
The minimum at a = 1. The density of the smallest coordinate at s > 0 is
e^(-s) (Z_{n-1}(2)/Z_n(1)) E_{n-1,2}[ 1_{r_i > s for all i}
prod over i of (1 - s/r_i)^2/(1 + s/r_i) ],
and the normaliser ratio is exactly n(n+1)/2. With
q_s(r) = 1_{r > s}(1 - s/r)^2/(1 + s/r), one has 0 <= q_s(r) <= 1 and
1 - q_s(r) <= 3s/r for all r > 0, so
prod over i of q_s(r_i) >= 1 - 3s sum over i of r_i^(-1).
The inverse-moment bound at a = 2 makes the expectation at least 1/2 for
0 < s < eps n^(-2) with eps small and fixed. The density is then bounded
below by a positive multiple of n^2, and an interval with endpoints fixed
positive multiples of n^(-2) has probability bounded below, on which
r_1^(-1/4) ~> n^(1/2). This gives the third display; n = 1 is direct. End of
proof.
Completion. Compare the positive law with the reference laws on both sides
of the inverse comparison.
Upper estimate. Condition the target law on y_N < 1/2. This decreasing event
shifts all coordinates down and increases the inverse statistic. On the
shorter interval compare with y_i = r_i/(bN) under the Bures-Laguerre law
with a = 1/2, conditioned on r_N < bN/2. Up to a constant the
target-to-reference likelihood ratio is the product of omega(y) y^(1/2)
e^(bNy), and
d/dy log(omega(y) y^(1/2))
= (1 - 2y - sqrt(2y/(1+y)))/(4y(1 - y)) >= -4, 0 < y <= 1/2,
the numerator being positive for y <= 1/8 and at least -1 on the rest. The
likelihood factors thus increase for every N >= 1 once b >= 4, and by
association the conditioned reference has the larger inverse statistic.
Choosing b/2 >= K_{1/2} keeps its conditioning probability bounded below,
so
E ( sum over i of y_i^(-1) )^(1/4)
~< (bN)^(1/4) E_{N,1/2} ( sum over i of r_i^(-1) )^(1/4)
~< N^(3/4).
Lower estimate. Set y_i = 2 z_i/(1 + z_i), with 0 < z_1 < ... < z_N < 1. The
transformed pair kernel is
K(2z_i/(1+z_i), 2z_j/(1+z_j))
= 2 K(z_i,z_j) /
( (1+z_i)(1+z_j)(1 + 2 z_i z_j/(z_i + z_j)) ),
so with the Jacobian the one-variable factors are
omega(2z/(1+z))(1+z)^(-(N+1)) and the extra pair factors are
(1 + 2 z_i z_j/(z_i + z_j))^(-1). Compare with z_i = r_i/(bN) under the
reference law with a = 1 and fixed 0 < b < 1/2, conditioned on r_N < bN. The
individual factors
omega(2z/(1+z)) (1+z)^(-(N+1)) e^(bNz)
and the extra pair factors are each decreasing in every coordinate, since
omega is decreasing and -(N+1)/(1+z) + bN < 0 for 0 < z < 1. Association
puts the target z coordinates below those of this conditioned reference,
themselves below the unconditioned reference. As y_1 <= 2 z_1,
E y_1^(-1/4) >= 2^(-1/4) (bN)^(1/4) E_{N,1} r_1^(-1/4) ~> N^(3/4),
and the inverse comparison gives m_N ~ N^(3/4).
Bridge estimate. Summing the lower increment bound over 1 <= j < N by
monotonicity gives (N-1) B_N ~< m_N, so B_N ~< N^(-1/4), with bounded N
absorbed. Conversely choose a fixed integer L with the lower bound for m_{LN}
twice the upper bound for m_N. Then
N^(3/4) ~< m_{LN} - m_N ~< sum over j = N to LN-1 of B_j
<= (L - 1) N B_N,
so B_N ~> N^(-1/4). This, with m_N ~ N^(3/4), c A_N + B_N = 1, and the
increment comparability, is the theorem. End of proof.
The four local identities are now verified coefficient by coefficient, which
completes the only deferred step.
Factorization. R(s lambda) = S_s S_s^T follows by direct substitution from
the weights: for s = 2 the cup coefficient is rho and each occupied strand
passes with coefficient rho, and for s = 3 the flat row R(3 lambda) passes
each route straight through while S_3 S_3^T reproduces the empty diagram
plus the cup.
Splitting. Subscripts 1, 2 mean the arguments x + s lambda and x. Let q be
the cup coefficient and k the strand coefficient of the splitter, so
q = k = rho for s = 2 and q = 1, k = 0 for s = 3. Unsubscripted weights are
those of Rhat_s; in the deletion case s = 3 use B = 1 and set its other
nonempty two-slot coefficients to zero. Inputs are the passing auxiliary
strand and the slot to be split (absent in the deletion case); outputs 1, 2
are the split slots and output 3 the auxiliary slot. A bare output index is
the single strand's destination and a parenthesised pair marks an extra cup.
Comparing R_2(x) R_1(x + s lambda)(I (x) S_s) with (S_s (x) I) Rhat_s:
input output connection left side right side
00 empty 1 1
00 12 (cup) U_1 A_2 + q B_1 B_2 q
00 13 U_1 B_2 + q B_1 A_2 k U
00 23 U_2 + q A_1 E_2 k U
10 1 A_1 + q F_1 U_2 k A
10 2 B_1 A_2 + q U_1 B_2 k A
10 3 B_1 B_2 + q U_1 A_2 B
10 3,(12) q F_1 E_2 q B
10 1,(23) A_1 U_2 + q(E_1 E_2 + F_1 F_2) 0
For s = 2 there are also, with a bare output pair denoting two
through-strands and an empty or parenthesised output joining the two input
strands:
01 1 k(B_1 + U_1 U_2) k B
01 2 k(A_1 A_2 + B_2) k B
01 3 k(A_1 B_2 + A_2) A
01 1,(23) k(B_1 U_2 + U_1 F_2) 0
01 3,(12) k U_1 E_2 q A
11 empty k(U_1 + B_1 U_2) U
11 12 k(E_1 A_2 + A_1 B_2) 0
11 (12) k F_1 A_2 q U
11 13 k(E_1 B_2 + A_1 A_2) k E
11 (13) k F_1 B_2 k F
11 23 k B_1 E_2 k E
11 (23) k(U_1 U_2 + B_1 F_2) k F
The tables follow from substitution. The weights at x + j lambda have the
common denominator s_{j+2} s_{j+3}, with s_j = sin(x + j lambda), and
numerators
weight A B U E F
numer. d s_{j+5} s_j s_{j+5} d s_j s_{j+5} s_{j+6} -s_{j+7} s_j
together with the product-to-sum identities
s_{j+8} = -s_j, s_j s_{j+1} + s_{j+4} s_{j+5} = C,
s_2 s_6 = d^2 - s_0^2, s_1 s_5 = d^2 - s_7^2,
rho(s_1 - s_7) = s_0, rho(s_6 - s_0) = s_7,
rho(s_1 s_2 - s_6 s_7) = s_0 s_4, rho(s_5 s_6 - s_0 s_1) = s_7 s_3,
rho(s_7 s_0 + s_4 s_5) = s_6 s_2, rho(s_2 s_3 + s_7 s_0) = s_1 s_5,
s_1 s_6 - s_7 s_0 = rho d^2,
s_1 s_6 s_5 = s_7 s_2 s_3 + d^2 s_0, s_1 s_6 s_2 = s_0 s_4 s_5 + d^2 s_7.
The last two follow from the preceding s_1 s_5 and s_6 s_2, and
substitution verifies every entry.
Unitarity. The one-strand sector is multiplication of two matrices with
diagonal entries A and off-diagonal entries B, and the even sector needs
E(x) E(-x) = 1, U(x) + E(x) U(-x) = 0,
E(x) F(-x) + F(x) E(-x) + U(x) U(-x) = 0,
each following from the weights, a cap composed with a cup contributing a
loop and so vanishing.
Braid. First set y = s lambda with s = 2 or 3. Factorization and splitting,
with their transposes and the reversal of slot order, make both sides of the
braid relation equal to (I (x) S_s) Rhat_s(x) (S_s^T (x) I), the effective
box being symmetric. Unitarity transports these identities to x + y = s
lambda, applying the identity with arguments (-x, x + y), and y = 0 follows
from R(0) = I.
For the remaining parameters multiply the difference of the two sides of the
braid relation by their three scalar denominators and call the resulting
diagram sum F(x, y). Each coefficient is a Laurent polynomial in e^(i y)
with exponents between -4 and 4. Let D_j multiply a diagram by -1 when slot
j is occupied; inspecting the nine boxes gives
R_i(v + pi) = D_{i+1} R_i(v) D_i.
On either side of the braid relation the two internal D_2 signs cancel,
leaving signs at input slot 1 and output slot 3, while the scalar
denominators are pi-periodic, so
F(x, y + pi) = D_3 F(x, y) D_1.
Every boundary coefficient has fixed parity, so after multiplication by a
Laurent monomial it is a polynomial of degree at most four in e^(2i y). It
vanishes at y = 0, 2 lambda, 3 lambda, 2 lambda - x, 3 lambda - x, which are
distinct modulo pi for generic x, so interpolation proves the identity and
rational continuation removes the genericity restriction.
This completes the proof of the four local identities, hence of the strip
power with uniform constants.