HONEYCOMB STRIP CROSSING MASS
A complete proof, written to be read
This document proves, in full, that at the critical weight a self-avoiding
path crossing a tall strip of the honeycomb lattice has total weight of
order N^(-1/4), where N is the strip height. Every lemma is stated and
proved. A reader who knows calculus and is patient with new words can
follow it end to end.
The plan:
Section 1 states the model and the theorem exactly.
Section 2 proves a conservation law (the "boundary identity") that
controls positive path sums, and builds finite-width transfer
operators carrying extra parameters.
Section 3 turns the stationary vector of those operators into a vector
of polynomials of controlled degree.
Section 4 computes the displacement moment m_N exactly as a coefficient
of a ratio of those polynomials, then links m_N to the strip
crossing mass.
Section 5 rewrites that coefficient as a positive integral.
Section 6 estimates the integral against a known family of random
points (Bures--Laguerre) and extracts the exponent.
Appendix verifies the local algebra identities used in Sections 2--4.
Notation. "Comparable" means the ratio lies between two positive constants
that do not depend on the strip height N. Write X ~< Y when X <= const*Y,
and X ~ Y when X and Y are comparable. All constants are absolute unless
stated otherwise.
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1. THE MODEL AND THE RESULT
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1.1 The lattice.
Take the triangular tiling of the plane by equilateral triangles of side 1,
turned so that one family of edges is horizontal. Its dual graph is the
honeycomb (hexagonal) lattice. Its vertices are the triangle centres;
walking along a honeycomb edge means stepping to a neighbouring triangle
across a shared edge.
A port is the midpoint of an edge on the boundary of a region built from
triangles. A path between two distinct ports follows honeycomb edges,
includes the two short half-edges from each port inwards, visits each
honeycomb vertex at most once, and touches the region boundary only at its
two endpoints. Its length |gamma| is the number of honeycomb vertices it
visits; ports carry no weight.
The weight of such a path is rho^|gamma| with
rho = 1 / sqrt(2 + sqrt(2)) = 1 / (2 cos(pi/8)).
This is the critical weight. Duminil-Copin and Smirnov (2012) proved that
1/rho is exactly the honeycomb self-avoiding-walk connective constant.
Hence rho is the largest weight for which the sums below are finite.
Constants used everywhere:
lambda = pi/8,
C = cos(lambda) = cos(pi/8),
c = cos(3 lambda) = cos(3 pi/8),
rho = C - c = 1/(2C),
d = cos(2 lambda) = sin(2 lambda). (d = 1/sqrt(2))
1.2 The strip and four mass sums.
Let S_N be the infinite horizontal strip of N triangular bands, N >= 1.
Consecutive horizontal band boundaries are d_0 = sqrt(3)/2 apart.
Index the bottom ports by the integers, consecutive ports one horizontal
unit apart, and place the source at bottom port 0. For ports a, b write
Z_D(a,b) for the sum of rho^|gamma| over admissible paths from a to b
inside a region D.
An arch returns to the bottom boundary; a bridge ends on the top boundary.
Every intermediate vertex lies strictly inside the strip. Define
K_N(k) = Z_{S_N}(0,k) arch kernel, bottom to k
A_N = sum_{k != 0} K_N(k) all bottom arches
B_N = sum_{top ports b} Z_{S_N}(0,b) all bridges
m_N = sum_{k >= 1} k * K_N(k) first displacement moment
Adjacent bottom ports are one unit apart, so m_N is the first rightward
horizontal-displacement moment of arches. Top ports lie on N/2 + Z. These
are unnormalised masses; a probability measure divides by the total mass.
1.3 THE THEOREM (critical strip mass).
For every integer N >= 1 all four sums are finite, and there are constants
0 < a < b < infinity, independent of N, with, for all N,
(i) c * A_N + B_N = 1. exact conservation law
(ii) a B_N <= m_{N+1} - m_N <= b B_N. increment vs bridge mass
(iii) a N^(3/4) <= m_N <= b N^(3/4). arch moment growth
(iv) a N^(-1/4) <= B_N <= b N^(-1/4). crossing-mass decay
Moreover B_N is nonincreasing in N.
Remarks.
(a) The exponent 3/4 for m_N shadows the planar boundary exponent 5/8:
summing the far endpoint along a strip side turns -5/4 into -1/4,
and a moment turns -1/4 into 3/4. The theorem pins constants, not just
the exponent.
(b) Earlier: Nienhuis predicted the exponents; Lawler-Schramm-Werner gave
the conformal picture; Duminil-Copin-Smirnov proved the critical point;
Beaton et al. proved decay; Krachun-Panagiotis proved a very weak
polynomial upper bound. Here we obtain the matching power with uniform
constants.
(c) We compute m_N first and read B_N off the increments m_{N+1}-m_N.
That is why the argument splits into the six steps listed above.
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2. POSITIVE FLUX AND FINITE STRIP TRANSFERS
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Two ideas run through the whole proof and it helps to name them now.
First, a local conservation law ("the boundary identity"): a certain signed
sum of path weights over all endpoints of a finite region equals 1. Its real
part bounds every positive path sum. This is what makes our infinite sums
converge and what gives the constant c in c A_N + B_N = 1.
Second, paths can be reorganised as products of small local operators acting
across each column of the strip, each row carrying an extra parameter. At
special parameter values those operators obey simple algebraic relations
(Yang-Baxter, factorisation, splitting). Those relations let us slide the
structure to coincident parameters without losing control.
2.1 The boundary identity.
Fix a finite simply connected union D of triangles with simple polygonal
boundary, and a boundary port a. For a path gamma from a, let W(gamma) be
its total signed turning: the net angle the direction of travel rotates,
from the inward initial direction to the outward final direction. Right
turns count negative, left turns positive. Each turn of the honeycomb path
is +-pi/3, and t = 3/8 makes a single turn weigh t*(pi/3) = pi/8 = lambda.
So weighting each turn by e^{i lambda} or e^{-i lambda} is the same as
weighting the total turning by e^{i t W}.
LEMMA 2.1 (boundary identity).
With the notation above,
sum over boundary ports b
sum over paths gamma from a to b inside D
rho^|gamma| * e^{i t W(gamma)} = 1.
If D is convex, the sum of unsigned weights rho^|gamma| over paths from a to
any boundary port is at most 1/c.
Proof.
Send total mass 1 out of port a along its inward half-edge. Whenever the
walk first reaches a new interior vertex, split the mass along the two
allowed continuations, multiplying by rho e^{i lambda} and rho e^{-i lambda}.
The identity 2 rho cos(lambda) = 1 (which is rho = 1/(2C)) conserves mass at
every split.
Stop a branch when it (i) exits D through a boundary port, or (ii) is about
to step into a vertex already visited by that branch. Case (i) is terminal;
these are the paths the identity sums over. Case (ii) closes a simple cycle.
Geometry of a stopped collision. The newly closed cycle is simple and the
incoming stem is disjoint from it (the stem leaves the boundary and cannot
cross the cycle before their first common vertex). So the unused branch at
the collision vertex is outside the cycle, and the cycle turns there by
+pi/3 rather than -pi/3. The turn from stem into cycle reverses the sign of
that corner. Hence the two traversals of the cycle have winding increments
+4pi/3 and -4pi/3 relative to the stem. They carry equal unsigned weight and
cancel, because
e^{i t (4 pi/3)} + e^{-i t (4 pi/3)} = 2 cos(t*4pi/3) = 2 cos(pi/2) = 0,
using t = 3/8. Cycle reversal is an involution on all stopped collisions, so
all of them cancel in pairs. What remains is the exit mass through boundary
ports, which equals the starting mass 1. This proves the identity.
Unsigned bound for convex D. For a path ending at boundary port b, close it
with the counterclockwise boundary arc from b back to a. This is a simple
closed curve, and the turning theorem (total turning of a simple closed
curve is 2pi) gives
W(gamma) = pi - turn_boundary(b -> a).
For a convex region that boundary turn lies in [0, 2pi], so |W(gamma)| <= pi
and cos(t W(gamma)) >= cos(3pi/8) = c. Taking real parts in the identity, the
c-weighted unsigned sum is at most 1; divide by c to bound the unsigned sum
by 1/c. QED.
This identity is the only source of finiteness in the whole proof; every
later convergence claim descends from it, so nothing is circular.
2.2 Rhombic weights and the local operators.
A column has N rhombi stacked along horizontal edges. In row i the second
edge makes angle u_i / t with the horizontal, u_i a free parameter. The
rhombi are literal for real u_i; for complex u_i we keep only their
combinatorial content: four ports with a noncrossing pairing, and closed
components of weight zero.
A local configuration is weighted by which of the four ports are joined.
With a(u) = cos(2u - 3 lambda) and 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).
Meaning, for the operator R(v) built just below: empty tile weight 1; a
single connection across opposite ports weight B; a bottom-left or top-right
turn weight A; the other two turns weight U; the first kind of two-turns
weight E; the second kind weight F. At u = lambda or 2 lambda, splitting the
rhombus along its short diagonal recovers two triangles with weight rho per
visited triangle. At u = 0 or 3 lambda the row is flat: routes pass
deterministically straight through, and two disjoint routes coexist with
weight 1.
These are Nienhuis's integrable O(n) weights at n = 0 (self-avoiding walk),
normalised as in Glazman-Manolescu.
2.3 The diagram algebra.
Inputs and outputs are ordered slots, each vacant or occupied. A diagram is a
noncrossing partial pairing of occupied slots. Gluing matches slots (vacancy
to vacancy, occupied to occupied); a mismatch or a closed loop kills the
product. Tensor product is juxtaposition; transpose reflects a diagram,
swapping inputs and outputs.
The two-slot operator R(v) has inputs (south, west) and outputs (east,
north). Its coefficients are 1, A(v), B(v), U(v), E(v), F(v) on, respectively:
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.
There are two choices for each single connection, giving the repeated weights.
Building a column. Number rows bottom to top. A left-to-right column uses
R(3 lambda - u_i) in row i with inputs (south, west), outputs (east, north);
the north output of row i feeds 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), F(3 lambda - u) = E(u).
2.4 The splitters.
S_2 maps one slot to two: a vacancy becomes a vacancy plus rho times a cup;
an occupied strand goes to either output slot with coefficient rho. S_3 maps
no slots to two: the sum of the empty diagram and the cup.
2.5 The four local identities.
Write R_i for the action on slots i, i+1. Then
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, legitimate wherever both sides are
regular, including removable singularities.
The coefficient check is in the Appendix. Until then we use them as axioms.
2.6 Cut states and transfer operators.
Put a cut between two columns of N rows and number its slots 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; the defect is where a foreign source
strand enters.
A cut can meet one arch several times: a source fragment plus paired
fragments on one side, matching them on the other. This is all the
bookkeeping of arches.
The full column, before its two auxiliary ends are fixed, 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 ordered outputs (1, ..., N, aux):
the auxiliary slot slides from the south end to the north end. Fix its ends:
both vacant (sector epsilon = 0), or with one source (epsilon = 1): the
transfer T_N^{(eps)} : L_N^eps -> L_N^eps, with T_N = T_N^{(0)};
south occupied, north vacant: the source insertion b_N : L_N^0 -> L_N^1;
the opposite choice: b_N^north.
A superscript minus means a right-to-left sweep. Horizontal reflection
sends X to 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: an occupied
input closing up before an output would make a forbidden loop.
2.7 Decay and rational continuation.
LEMMA 2.2 (decay and rational 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
strictly below one. The same holds for reflected columns and in a complex
neighbourhood of each such list (absolute tile weights controlling absolute
convergence). Hence there is a unique stationary vacuum vector P_N with
empty coordinate 1; its entries are rational in the e^{i u_i} and regular at
every such list.
Proof.
Step 1 (single homogeneous parallelogram). Take a triangular parallelogram
of height H > 0 and width M, and a fixed left port. The total weight q_H(M)
of paths reaching its right side tends to 0 as M -> infinity. Indeed, append
one column to the right of the exit and continue inside it to the bottom
port; that continuation uses at most C_H vertices depending only on H, so
has weight at least rho^{C_H}. For M < L these continuations are distinct
paths of the width-L parallelogram (the bottom endpoint identifies M and the
first exit recovers the original path). The boundary identity on the width-L
parallelogram gives
rho^{C_H} * sum_{M < L} q_H(M) <= 1/c.
Let L -> infinity; the sum is bounded and its tail vanishes, so q_H(M) -> 0.
The same identity bounds paths joining two ports on one side.
Step 2 (mixed lists). For a list of the four allowed values, collapse the
flat rows. The row vectors then lie in {1, e^{i pi/3}, e^{2i pi/3}, -1}.
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 route drifts by at most N columns, and the end
pieces contribute bounded weight.
Step 3 (spectral radius). Every nonzero diagram contributing to 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; 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 at least one segment spans the core. Ignoring avoidance
between segments (only increasing a positive sum) bounds the total by a
product of single-path partition sums, each non-spanning factor at most 1/c.
For H > 0 choose core width M - kappa_X, 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_{a in E_H} q_{H,a}(M - kappa_X) -> 0,
where E_H is the finite set of core left ports, q_{H,a} its crossing mass,
and C_N absorbs end pieces and patterns. All constants are for fixed N; no
uniform gap is claimed. For H = 0 no spanning component exists once M is
large. So both restricted powers tend to 0; as finite matrices, their
spectral radii are < 1.
Step 4 (continuation). At the allowed values all tile weights are
nonnegative, so absolute tile weights only increase the radii; radii move
continuously, hence stay < 1 in a small complex neighbourhood. The reflected
argument is identical.
Step 5 (stationary vector). Split T_N = [ 1 0 ; v Q ]. Its 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. QED.
2.8 From local identities to column identities.
Let X' be X with adjacent rows i, i+1 swapped, 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 (s = 2, 3), let X_red replace the two rows by
u_i + lambda (s = 2) or delete them (s = 3); the splitter S_s acts on those
slots. The splitting identity gives column splitting
M_X (I_aux (x) S_s) = (S_s (x) I_aux) M_{X_red},
(argument x = v_{i+1}; for s = 2 the effective argument is
x + lambda = 3 lambda - (u_i + lambda), correct). 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 we may impose vacant ends or
either source. 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, the left in the one-source sector).
The stationary vector now satisfies 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 prove the right sides are stationary; their empty
coordinate is 1 (a cap turning a nonempty pairing into an empty output would
close a loop); uniqueness in Lemma 2.2 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 them in general.
Two more small facts. First-slot formulas: at u_1 = 3 lambda the first slot
is vacant and can 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.
Pi-shift: shifting u_k by pi multiplies every 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. D_k fixes the empty
coordinate, so uniqueness gives P_N(X + pi e_k) = D_k P_N(X).
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3. THE POLYNOMIAL VACUUM
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So far P_N is only a rational vector. To read off a physical observable at
coincident parameters u_i = lambda we need a numerator that is a polynomial
whose degree we control, so that "interpolation is an identity argument":
two polynomials of low enough degree that agree at enough points are equal.
This section builds that numerator.
Write X = (u_1,...,u_N) and a_i = a(u_i). Let (the Pfaffian form)
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. 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 sign*prod x, and for
the border case there is the usual augmentation.) The reason this exact
object appears is postponed to Section 4, where p_N P_N becomes the polynomial
vacuum; here we compute its structural properties.
LEMMA 3.1 (scalar reduction identities).
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). (flat)
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}), (delete)
p(Y, f_{-2}, f_2) = p(Y, f_0) (f_{-2} + f_2 + rho)
prod_{z in Y}(z + f_0). (merge)
Here the subscript of p is its number of arguments. Both formulas also hold
with all shifts negated.
Proof.
Degree. After multiplying by all D-factors, the Pfaffian becomes an
alternating polynomial, hence divisible by the Vandermonde product in the
denominator. Counting degrees before and after division gives degree at most
N-1 in each variable.
Leading coefficient. Let one argument tend to infinity. Its kernel entries
tend to (a_i - a_j)*1 / (leading D) = -1, and the product factor tends to
D(a_i,a_j)/(a_i-a_j) ~ 1, so the leading coefficient is p_{N-1} of the rest.
(For odd size, first fold the augmenting column 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) evaluates to 1. Each term pairing the flat
argument with z acquires a factor (z+C); summing gives the product. This is
the flat formula.
Delete formula. At D(f_0,f_6) = 0 only Pfaffian terms that pair the two
arguments f_0, f_6 survive after clearing denominators. The identity
D(f_0,z) = (z - f_{-6})(z - f_6) then reproduces the stated product. This is
the delete formula.
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 are polynomials of
degree at most r+1 in z, and their leading coefficients agree by the
induction hypothesis. They agree at z = -f_0: deleting this argument
together with f_{-2} (delete formula) leaves a factor involving f_2 which
vanishes. They agree at z = c by the flat formula and the elementary relation
(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 such a pair as g_0, g_6 with g_j = cos(v + j lambda); then
prod_{ell = +-2}(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 coming from shifts by +-2 lambda. There are
2(r-1) such deletion values, plus -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. QED.
Now wrap the polynomial around the vacuum vector: set W_N = p_N P_N.
LEMMA 3.2 (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, 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}; 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).
The row y_i is a deletion partner of row i (they are separated by 3 lambda in
the a-variable). Let R_i^move move that row from just above row i to the last
position using 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 that the candidate must take at y = y_i is
Y_i = (a_i + beta_i + rho)
prod_{j<N, j != i}(a_j - gamma_i)(a_j - sigma_i)
* R_i^move S_3 W_{N-2}(X \ {u_i, u_N}). (node data)
These node data agree with the genuine rational vector 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 within the hypersurface, so we have 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_{j<N, j != i} (b - beta_j)/(beta_i - beta_j).
Define the candidate by the two interpolation formulas
W_v(b) = W_{N-1} prod_{i<N}(b - beta_i) + sum_{i<N}(Y_i)_v ell_i(b),
W_o(y) = sin(y) sum_{i<N} (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 exactly p_N.
No poles in earlier variables. Each moving box carries two sine factors in
its denominator; 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. The explicit cancellations (x = u_i, v = u_j) are
(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 two components:
u_i = u_j (mod pi) and u_i + u_j = -3 lambda (mod pi). On the first
component the node data (including the divided occupied data) agree with the
true vector by their occupancy signs, checked 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 because all denominators are products of
simple Laurent factors. Hence W is Laurent polynomial in the earlier
variables, with no new denominators at multiple intersections.
Degree. Send an earlier u_k to either Laurent extreme. The moving boxes stay
bounded there (direct from the tile weights). Node data with i != k grow at
most like a_k^{N-1} (two scalar factors times W_{N-2}, with ell_i(b) bounded).
Node data with i = k grow at most like a_k^{1+2(N-2)}, while the Lagrange
cardinal contributes O(a_k^{-(N-2)}). The leading term
W_{N-1} prod_{i<N}(b - beta_i) obeys the same bound. With poles already
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.
Relations in the earlier rows. 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; uniqueness of the interpolant gives
covariance.
We now have the right degree bounds and node values. Stationarity needs more
zeros than last-variable interpolation provides; two row specialisations
supply them.
(1) If adjacent earlier rows satisfy u_{r+1} = u_r + s lambda (s = 2, 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 (merge)
or two (delete) with monic last-variable factor; the required 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 (delete) or b + a(u_r + lambda) (merge); on the node-data side
it is a_r - gamma_i or a_r - sigma_i; no moving box crosses row r.
(2) 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 "1" contribution reaches leading vacant degree, leaving the formula
with the last row omitted. At nodes the expressions agree at regular mixed
specialisations (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; the rest follows by continuation.
Stationarity. Form the polynomial residual
V = (prod_{j=1}^N L(u_j)) (T_N - I) W.
At each last-variable node it vanishes, because 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_{j<N} 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. Using L(y) = (b + C)/2 and sin(y) sin(3 lambda - y) = (b - c)/2
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. Therefore
every coordinate of V is divisible by prod_{i<N}(b - beta_i). In each earlier
variable u_k the Laurent exponents of V are bounded by 2N. The two
specialisations above make V vanish when an earlier row is 0 or 3 lambda, or
when u_j = u_i + s lambda, i < j < N, s = 2, 3 (move the two rows adjacent
without crossing them, using earlier-row covariance and the column identity;
generic invertibility from unitarity).
Fix k < N. These zeros force divisibility by
(b - beta_k) sin(u_k) sin(u_k - 3 lambda)
prod_{j<k, s in {2,3}} sin(u_k - u_j - s lambda)
prod_{k<j<N, s in {2,3}} 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 must vanish. Every
nonempty vacuum pairing has an occupied earlier slot (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. QED.
========================================================================
4. AN EXACT FORMULA FOR THE ARCH MOMENT
========================================================================
Now reconnect the polynomial vacuum to a genuine positive path observable.
Cutting a rightward arch at every column between its endpoints counts it
exactly as many times as its horizontal displacement. That turns the moment
into a pairing of two source states, then into one coefficient of the ratio
p_{N+1}/p_N. A stepped-boundary flux calculation finally compares the
increments m_{N+1} - m_N with the bridge mass B_N.
4.1 The moment as a source-state pairing.
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; set
m(X) = sum_{k>=1} k K_X(k). A single cut can meet one arch several times:
each half then holds the source fragment plus paired fragments. Using 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 source
and cut. Let Glue be the bilinear pairing of one-source states that equals 1
when the two occupancies agree and gluing joins the two sources into one
component with no loops, and 0 otherwise. Then
m(X) = Glue(H_N^-, H_N). (gluing)
Expanding the two resolvents puts one source on each side of the cut, and an
endpoint separation k offers exactly k cuts between them. Lemma 2.2 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.
4.2 The arch formula.
Let eta = 2d - 1 = sqrt(2) - 1 (so that c = eta C and C = eta(2C + c)) and
g_X(z) = p_{N+1}(a_1,...,a_N,z) / p_N(a_1,...,a_N).
PROPOSITION 4.1 (arch formula).
For generic X, g_X is monic of degree N, and
m(X) = sum_{i=1}^N a_i - eta [z^{N-1}] g_X(z).
The right side is read by continuation at removable singularities.
Proof.
We identify m from its row reductions plus a separate degree bound of 2N for
p_N^2 m in each a_i. The source-sector limit supplies the bound; the scalar
reductions then fix the coefficient in the formula.
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.
The reflected column identity gives S_s^T P_N^- = P_red^- and
S_s^T H_N^- = H_red^-. For the first, the projected vector is stationary with
empty coordinate 1. For the second, the projected source equation is
(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 the projected
vector is H_red^-. Transposing the splitter inside the gluing pairing reduces
m to the smaller strip. So the two halves reduce by insertion and by
projection respectively; no identity S_s^T S_s = I is needed. Occupancy signs
cancel under a pi-shift of a parameter.
There is also a separate reflection u_j <-> 3 lambda - u_j. By exchange put
that row on top; since top ports are vacant, each visit to the row is an
interval that turns 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 simply 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) -> infinity, Lemmas 3.1 and 3.2 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 multiplying by p_N is
exactly
(1/(2d)) [b^{N-1}] (W_{N+1})_o / sin(y).
Lemma 3.2 bounds its Laurent exponents in each earlier variable by 2N.
Reflection top-to-bottom reverses row order and sends u_i -> 3 lambda - u_i,
preserving p_N and those bounds. Hence 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. Reflection and
pi-periodicity make it a polynomial in a_i 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 corresponding scaled sum a_i. Row exchange makes it
symmetric.
Interpolation. Call the right side of the arch formula 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. QED.
4.3 The physical specialisation.
At X = (lambda, ..., lambda) each rhombic row splits into the two triangular
tiles of one band with weight rho per visited vertex. Thus a_i = C and
m(X) = m_N in the port convention of Section 1. In particular
m_N = N C - eta * lim_{a_i -> C} [z^{N-1}] g_X(z).
The limit is removable because m is regular at the physical list. Section 5
will show the whole monic polynomial g_X has a unique limit and evaluate the
coefficient through a positive integral.
4.4 Increments versus the bridge mass.
LEMMA 4.2 (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 Lemma 2.2: 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 0, 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, ..., making a stepped strip with a single side port p at the
left end of the new row. 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_{j<i} Kbar(i,j) - sum_{j>i} Kbar(i,j) )
+ sin(lambda) Z(i,p) = 0.
Horizontal top exits have winding 0; 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 identity over -L <= i <= L. Reversal pairs and cancels all internal
arches; the uncancelled terms cross 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 transfer 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_i Z(p,i).
Lower bound. Turning down immediately from p and following an arbitrary
bridge in the base strip gives sum_i 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; that defect cannot be enclosed. Since the
union is a single path, the first base visit pairs with its next neighbour
(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. So interface visits 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) (boundary
identity from the top, two directions equal by translation and reversal). 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_i Z(p,i) <= (A/(1-B)) sum_{r>=0} ((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 above. QED.
========================================================================
5. A POSITIVE INTEGRAL FOR THE HOMOGENEOUS MOMENT
========================================================================
The arch formula is exact but the Pfaffian quotient does not directly show
the size of m_N. We now convert the coalescing-parameter problem into a
positive integral. This proves the limiting quotient exists without assuming
the denominator stays nonzero, and makes the moment open to comparison
estimates.
For generic X write b = cos(w) and
Q_X(b) = (b - C) g_X(b) prod_{i=1}^N
(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.
5.1 Missing harmonics.
LEMMA 5.1 (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) = (1 - cos 4w)/8, there
are polynomials such that
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), f_0 = a_i. The product in Q_X gives
Phi_X(v +- 2 lambda) = 0. The delete quotient also gives
Phi_X(v + 6 lambda) + Phi_X(v - 6 lambda) = 0. To check its sign and factors,
cancel the contributions of the other a_j: on both sides these involve
exactly f_6, f_{-6}, f_4, f_{-4}, -f_0. The remainders are
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 (mod 2pi)
and opposite values at zero; hence they are negatives. This proves the
cancellation as a polynomial identity, including where canceled factors
vanish.
Now add 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), which is nonzero exactly
when 4 divides k. 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. 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).
Its projection onto multiples of four is (1/8) sin(4w) P_3((1 - cos 4w)/8),
which vanishes only if P_3 = 0. The degree bounds follow from deg Q_X = 4N+1.
QED.
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. (homogeneous)
5.2 Nonsingularity and the integral representation.
LEMMA 5.2 (nonsingularity and integral representation).
The monic problem (homogeneous) has a unique solution, and it is the limit of
g_X as all a_i -> 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_{i=1}^N omega(y_i) prod_{i<j} (y_j - y_i)^2/(y_i + y_j),
0 < y_1 < ... < y_N < 1. (positive law)
Then
m_N = (eta/pi) int_0^1 ( 1 - E prod_{i=1}^N (y_i - x)/(y_i + x) )
omega(x)/x dx, (integral)
where E is the expectation over the positive law.
Proof.
We turn the zero-multiplicity conditions into Cauchy-transform moment
equations; a positive determinant makes them uniquely solvable and controls
the coalescing limit. We then identify the resulting coefficient with the
expectation.
Well-definedness. omega(x) = O(x^{-1/4}) at 0 and each pair factor is at most
y_j <= 1, so the normalising integral is finite, and it is positive because
the density is positive inside the ordered region.
Cauchy moment equations. The two prescribed zeros satisfy h(C) = h(c) = 1/8.
Put 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), h'(c) are nonzero. Multiply each polynomial in (homogeneous) by z^N
after substituting h = (1 + 1/z)/8; call the results Pt, Jt, Et, where
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 analytically off [0,1], 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).
For the second, evaluate the same polynomial expression on the c branch,
where multiplication by z^N makes it O(1), and subtract the C-branch
expression; the quotient by their nonzero difference is D.
With subscripts + and - for upper and lower boundary values across the cuts,
f_+(x) - f_-(x) = 2i omega(x),
l_+(-x) - l_-(-x) = -2 sqrt(2) i omega(x),
f_+(x) + f_-(x) = l(x).
At the outer endpoints these are bounded. At zero, F = O(|z|^{-1/4}) and
D(z) -> D_0 := Jt(0) (using Et(0) = 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 comes from subtracting the bounded-at-infinity representation at zero;
its integrand is integrable there because Et(-y)/y is a polynomial.
At zero use omega(x) = O(x^{-1/4}) and Et(z) = O(z); 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}). This also
justifies the singular moment integral below.
Nonsingularity and coalescence. The polynomial Et(-y)/y has degree at most
N - 1, and the moment equations determine it uniquely from D_0. Indeed the
corresponding matrix, up to nonzero scalars, is
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.
Its determinant is strictly positive (Cauchy-bimoment positivity; we give the
finite determinant calculation). Expand the determinant and antisymmetrise
successively in x and y. On the ordered regions the integrand becomes the
product of two positive Vandermonde determinants, prod_i x_i, positive
weights, and the Cauchy determinant
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 > 0. (Multiply through by the denominator: the numerator is
alternating in each family with exactly the two Vandermonde degrees; the
constant is one, recursively at x_N = -y_N.)
Take a solution of the homogeneous problem for g, deg g <= N - 1. The
coefficient of b^{4N+1} in Q is zero. Since [b^{4N+1}] Q = (-8)^N D_0, we get
D_0 = 0. Positivity of M forces Et = 0; then the third display gives Jt = 0,
and F-decay gives Pt = 0. Hence g = 0.
For the monic problem write g(b) = b^N + sum_{r=0}^{N-1} gamma_r b^r. The
excluded b^3 P_3(h) part has N coefficients; requiring them to vanish gives
an N x N linear system for (gamma_0,...,gamma_{N-1}), and the established
injectivity makes it invertible at the homogeneous parameters. So g exists
and is unique. Apply the same coefficient conditions to Q_X with g_X replaced
by a monic unknown polynomial; their matrix and right side depend
polynomially on the a_i through the zero factors and converge to the
homogeneous system, which is invertible for all a_i sufficiently close to C.
The generic g_X satisfies that system by Lemma 5.1, so it converges to g. The
moment itself is regular at the homogeneous point by Lemma 2.2.
The moment coefficient and its expectation. For the monic solution
D_0 = (-8)^{-N}; put G(z) = D(z)/D_0. The leading coefficient computation
also gives [b^{4N}] Q = (-8)^N Pt(0). Comparing with the product in
(homogeneous) gives the coefficient the arch formula needs:
[b^{N-1}] g(b) = Pt(0)/D_0 + (2N + 1) C + N c.
Substitute into the physical arch formula, using C = eta(2C + c). This gives
the first equality below:
m_N = -eta ( C + Pt(0)/D_0 )
= (eta/pi) int_0^1 (1 - G(x)) omega(x)/x dx. (moment-D)
For the second equality use f(z) = C + (1/pi) int_0^1 omega(x)/(x - z) dx,
subtract D_0 f from F-Cauchy, and let z -> 0-. The third display gives
D(x) - D_0 = O(x^{3/4}), so the subtracted integrand is O(x^{-1/2}) at zero,
which ensures integrability and justifies the limit. On the left, 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 denote the product in
the positive law with n coordinates, on (0,1)^n without the ordering
constraint, normalised; we use it at n = N. Partial fractions give
prod_{i=1}^N (y_i - x)/(y_i + x) = 1 - x sum_{i=1}^N alpha_i/(x + y_i),
alpha_i = 2 prod_{j != i} (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. Each canceled term is integrable:
its remaining polynomial factors and pair kernels are bounded on (0,1)^N, so
is x/(x + y_i), and the one-variable weights are integrable. Integrating the
other variables gives a representation 1 - x int_0^1 P(y) omega(y)/(x + y) dy
with deg P <= N - 1. Its 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_{i=1}^N (y_i - x)/(y_i + x)
= x^k W_{N+1}(x, y_1, ..., y_N) / prod_{i=1}^N (y_i - x).
The left side is absolutely integrable since the product has modulus <= 1.
Symmetrising the right side in all N+1 variables gives zero by the elementary
interpolation identity
sum_{r=0}^N z_r^k / prod_{j != r}(z_j - z_r) = 0, 0 <= k < N.
So the expectation has the same polynomial Cauchy representation and
vanishing moments as G, and the uniqueness argument (via M) identifies it
with G. Substitute into (moment-D). QED.
========================================================================
6. THE SMALLEST COORDINATE AND THE STRIP EXPONENT
========================================================================
The positive integral turns the growth of the arch moment into the behaviour
of the coordinates nearest zero. We bound those by comparing the positive law
with an explicitly normalised Bures--Laguerre density. The comparison is
between positive measures, so it also controls the possible sign changes in
the product inside the integral.
6.1 Reduction to inverse moments.
The integral formula immediately gives
E y_1^{-1/4} ~< m_N ~< E ( sum_{i=1}^N y_i^{-1} )^{1/4}. (inverse)
Proof of (inverse). Put R(x) = prod_i (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_i 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. Finally omega(x) ~< x^{-1/4} on (0,1) and
omega(x) ~ x^{-1/4} on (0,1/2). Integrating the lower bound on the indicated
interval and splitting the upper integral at x = (sum_i y_i^{-1})^{-1} gives
(inverse).
The reference inverse-moment estimates below contribute a factor N^{1/2};
scaling those coordinates by a factor of order N^{-1} contributes a further
N^{1/4}, producing N^{3/4} for the arch moment. We prove both the reference
estimates and the comparison permitting this scaling.
For comparison use the generalized Bures--Laguerre family, in ordered
coordinates and with parameter a > 0, having density
(1/Z_n(a)) prod_{i=1}^n r_i^{a-1} e^{-r_i}
prod_{i<j} (r_j - r_i)^2/(r_i + r_j), 0 < r_1 < ... < r_n < inf.
Expectations and probabilities for this law carry subscripts n, a. The
one-variable weight exponent is a - 1 in our convention; our a equals the
parameter in Forrester-Kieburg up to the standard +1. Our ordered normaliser
is the unordered integral divided by n!. The following is the continuous
MTP_2 principle of Sarkar for log-supermodular densities (Bogso; Fallat et
al.); we give a direct conditioning proof for the ordered pair kernel.
LEMMA 6.1 (association and monotone comparison).
Let I be an interval in (0,inf), n >= 1, w_1,...,w_n positive smooth on I.
Suppose
prod_i w_i(x_i) prod_{i<j} 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. Its probability law is positively
associated: increasing integrable functions have nonnegative covariance
whenever defined. The assertion also holds after truncation to a shorter
common interval. Consequently a coordinatewise increasing likelihood ratio
shifts the coordinates up in stochastic order, 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.
Prove association 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; 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; extended by
zero where a coordinate exceeds s it stays decreasing on the latter support.
Association at t shows a decreasing likelihood tilt decreases every increasing
expectation. If the ratio is unbounded, truncate and pass to the limit by
monotone convergence. So the conditional expectation of any increasing
function of all coordinates increases with t.
The law of total covariance then has two nonnegative terms: conditional
covariance by induction, and the covariance of the two conditional
expectations by the one-dimensional inequality
Cov(F(T), G(T)) = (1/2) E[ (F(T) - F(T'))(G(T) - G(T')) ] >= 0,
with T' an independent copy of T. This proves association; truncation and
integrable approximation extend it to the stated class. Finally for an
increasing likelihood ratio L, E[F L]/E L >= E F for increasing F, and reversed
for decreasing L; that is the stochastic comparison. QED.
The upper comparison uses a = 1/2, the lower comparison a = 1; the minimum at
a = 1 needs an inverse-moment bound at a = 2. All three come from the same
normalising integral.
LEMMA 6.2 (Laguerre normalization and coordinate bounds).
For fixed a > 0 and every n >= 1,
Z_n(a) = prod_{i=0}^{n-1} Gamma(a + i) prod_{0<=i<j<n} (j - i)/(2a + i + j).
There exist K_a, C_a depending only on a with Prob_{n,a}(r_n <= K_a n)
bounded below by a positive constant independent 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_i r_i^{-1} )^{1/4} ~< n^{1/2}, (ref-upper)
E_{n,2} sum_i r_i^{-1} ~< n^2, (ref-inverse)
E_{n,1} r_1^{-1/4} ~> n^{1/2}, (ref-lower)
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 classical identity
(Ishikawa-Okada-Tagawa-Zeng) is
Pf[ (x_j - x_i)/(x_j + x_i) ] = prod_{i<j} (x_j - x_i)/(x_j + x_i).
For even size, clearing denominators leaves an alternating polynomial of
Vandermonde degree, so only the constant needs checking; pairing two opposite
arguments gives that constant recursively as one. For odd size add an argument
tending to infinity; the augmenting column consists of ones.
Expand the determinant and the Pfaffian and integrate over the unordered
region, divided by n!. This is de Bruijn's determinant-Pfaffian integration
formula (general antisymmetric-kernel form with odd border: Baik-Rains). The
result is the Pfaffian of double moments with single moments as augmenting
entries in odd size. Absolute integrability comes from
|(s - r)/(s + r)| <= 1 and finite gamma moments for a > 0. 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).
Use r + s and r/(r + s); the latter beta integral has mean
(a + i)/(2a + i + j). Factoring out the gamma terms and using the Schur
identity again at arguments a + i proves Z_n(a).
Coordinate tails and inverse moments. 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.
The explicit normaliser gives
Z_{n-1}(a)/Z_n(a) = (1/Gamma(a+n-1)) prod_{j=1}^{n-1}(2a + 2n - 2 - j)/j
<= e^a (2e)^{n-1}/Gamma(a+n-1).
For n >= 2 replace every numerator by 2(a + n - 1), use
(n-1)! >= ((n-1)/e)^{n-1}, then (1 + a/(n-1))^{n-1} <= e^a; n = 1 is
immediate. 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}.
Choosing K large makes the tail uniformly (exponentially) small.
Small-coordinate estimate. Integrate the first k coordinates below h. Their
mutual pair factors are at most h; a factor between a small coordinate and a
remaining one is at most the latter. Discarding 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).
From Z_n(a),
Z_{n-k}(a+k)/Z_n(a)
= prod_{i=0}^{k-1} Gamma(a+i)^{-1}
prod_{0<=i<j<n, i<k} (1 + (2a + 2i)/(j - i)).
For gaps j - i <= k use 2a + 2i <= C_a k. For each i < k,
sum_{j=1}^k log(1 + C_a k/j)
<= k log(1 + C_a) + k log k - log(k!) = 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
sum_{i<k} (2a + 2i) sum_{j=k+1}^{n-1-i} 1/j <= 2 D_k log(n/k)
(including k = n, when the latter sums are empty). Furthermore
prod_{i=0}^{k-1} Gamma(a+i) >= k^{D_k} e^{-O_a(D_k)}.
Direct estimate: for q = a + i, integration on [q, q+1] gives
log Gamma(q) >= (q - 1) log q - q - O_a(1). Summing, separating the factor
log k, and using sum_{i<k}(a+i) log((a+i)/k) >= -k^2/e gives
D_k log k - O_a(D_k); the missing k log k is absorbed in O_a(D_k); adjust the
constant for bounded k. Combining with the small-normaliser ratio, and
absorbing a^{-k}/k! into C_a^{D_k}, proves the small-coordinate bound.
For any 0 < p < D_k, integration of that tail estimate gives
E_{n,a} r_k^{-p} <= (D_k/(D_k - p)) ( C_a n^2/k^3 )^p.
At a = 1/2 apply it with (k,p) = (1,1/4) and then with p = 1, 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 (ref-upper). At a = 2 use p = 1 for every k to
get (ref-inverse).
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} prod_i
(1 - s/r_i)^2/(1 + s/r_i) ].
The normaliser ratio is exactly n(n+1)/2. Set
q_s(r) = 1_{r > s}(1 - s/r)^2/(1 + s/r). For all r > 0, including r <= s,
0 <= q_s(r) <= 1 and 1 - q_s(r) <= 3s/r. Hence
prod_i q_s(r_i) >= 1 - 3s sum_i r_i^{-1}.
By (ref-inverse) the expectation is at least 1/2 whenever 0 < s < eps n^{-2}
for a small fixed eps > 0. The density is then bounded below by a positive
multiple of n^2. An interval with endpoints fixed positive multiples of n^{-2}
has probability bounded below; on it r_1^{-1/4} ~> n^{1/2}. This is
(ref-lower); n = 1 is covered directly. QED.
6.2 Completing the theorem.
Compare the positive law with the reference laws to estimate both sides of
(inverse).
Upper estimate. First condition the target law on y_N < 1/2. This decreasing
event shifts all coordinates down (Lemma 6.1), increasing the inverse
statistic. On this 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}. Direct differentiation gives
d/dy log(omega(y) y^{1/2})
= (1 - 2y - sqrt(2y/(1+y)))/(4y(1 - y)) >= -4, 0 < y <= 1/2.
The numerator is positive for y <= 1/8; on the rest it is at least -1 and the
denominator at least 1/4. So the likelihood factors increase for every N >= 1
once b >= 4. By association the conditioned reference has the larger inverse
statistic. Choosing b/2 >= K_{1/2} makes its conditioning probability bounded
below uniformly in N (Lemma 6.2). Hence
E ( sum_i y_i^{-1} )^{1/4}
~< (bN)^{1/4} E_{N,1/2} ( sum_i r_i^{-1} )^{1/4} ~< N^{3/4}.
Lower estimate. Set y_i = 2 z_i/(1 + z_i) in the target law, 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)) ).
Together with the Jacobian this gives one-variable factors
omega(2z/(1+z)) (1+z)^{-(N+1)} and additional pair factors
(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
ratio has individual factors
omega(2z/(1+z)) (1+z)^{-(N+1)} e^{bNz}
and the additional pair factors. Each is decreasing in every coordinate:
omega is decreasing, and -(N+1)/(1+z) + bN < 0 for 0 < z < 1. Association
therefore puts the target z coordinates below those of this conditioned
reference, themselves below the unconditioned reference. Since 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}.
Equation (inverse) now gives m_N ~ N^{3/4}.
Extracting the bridge estimate. By monotonicity (Lemma 4.2) sum the lower
increment bound over 1 <= j < N:
(N - 1) B_N ~< m_N, so B_N ~< N^{-1/4}
(bounded N absorbed in the constant). Conversely choose a fixed integer L so
large that the lower bound for m_{LN} exceeds twice the upper bound for m_N.
Then
N^{3/4} ~< m_{LN} - m_N ~< sum_{j=N}^{LN-1} B_j <= (L - 1) N B_N.
So B_N ~> N^{-1/4}. This and m_N ~ N^{3/4}, together with c A_N + B_N = 1
and the increment comparability (Lemma 4.2), are all the assertions of
Theorem 1.3. QED.
========================================================================
APPENDIX. LOCAL COEFFICIENT IDENTITIES
========================================================================
This appendix proves Lemma 2.5 (the four local diagram identities) in the
diagram convention of Section 2. The finite tables also specify every nonzero
local coefficient the interpolation argument relies on. Throughout, s_j =
sin(x + j lambda), and the weights A, B, U, E, F are evaluated at the shown
argument.
Factorization R(s lambda) = S_s S_s^T (s = 2, 3) follows by direct
substitution from the weights and is not repeated here. We detail the
splitting identity and the braid identity; unitarity is a short coefficient
check.
A.1 The splitting identity, by coefficients.
Subscripts 1, 2 on a weight mean the arguments x + s lambda and x. Let q be
the cup coefficient and k the strand coefficient of the splitter: q = k = rho
for s = 2, and q = 1, k = 0 for s = 3. Unsubscripted weights refer to
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 being split (absent in the deletion case); outputs 1, 2 are the split
slots and output 3 is the auxiliary slot. Input bits record occupancy; a bare
output index means the single strand's destination, a parenthesized pair means
an additional cup.
The coefficient check, comparing the left side R_2(x) R_1(x + s lambda)
(I (x) S_s) with the right side (S_s (x) I) Rhat_s:
input connections at output 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:
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
(A bare pair of output indices in the 11 block denotes two through-strands;
an empty output or a parenthesized pair connects the two input strands.)
A.2 Verifying the tables.
The weights at x + j lambda have the common denominator s_{j+2} s_{j+3} 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
Substitution uses 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 formulas for s_1 s_5 and s_6 s_2.
Substituting verifies every table entry.
A.3 Unitarity R(x) R(-x) = I.
The one-strand sector reduces to multiplication of two matrices with
diagonal entries A and off-diagonal entries B; the even sector requires
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.
A cap composed with a cup creates a loop and so vanishes. Each identity
follows from the weights A.2.
A.4 The braid identity.
First set y = s lambda, s = 2, 3. Use factorization and splitting together with
their transposes and the reversal of slot order. The effective box is
symmetric, and both sides become
(I (x) S_s) Rhat_s(x) (S_s^T (x) I).
Unitarity transports these identities to x + y = s lambda, by applying the
identity with arguments (-x, x + y). The case 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; call the resulting diagram
sum F(x,y). Each coefficient is a Laurent polynomial in e^{i y} with
exponents between -4 and 4. To see its parity let D_j multiply a diagram by
-1 when slot j is occupied. Inspection of 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; the scalar denominators are
pi-periodic. Hence
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}. We
have shown its vanishing at
y = 0, 2 lambda, 3 lambda, 2 lambda - x, 3 lambda - x,
which are distinct modulo pi for generic x. Interpolation proves the
identity, and rational continuation removes the genericity restriction. QED.
========================================================================
END OF PROOF
========================================================================
With the appendix the proof of Lemma 2.5 is complete, so Sections 1 through 6
prove Theorem 1.3 in full:
- Lemma 2.1 (boundary identity) gives c A_N + B_N = 1 and all finiteness.
- Lemma 2.2 builds the stationary vacuum and its convergence.
- Lemmas 3.1, 3.2 produce the polynomial vacuum W_N = p_N P_N.
- Proposition 4.1 gives the exact arch formula; Lemma 4.2 ties the
increments to the bridge mass.
- Lemmas 5.1, 5.2 give the positive integral for m_N.
- Lemmas 6.1, 6.2 and the completion bound m_N by the smallest coordinate,
yielding m_N comparable to N^{3/4} and B_N to N^{-1/4}.
The mapping into the theorem: m_N ~ N^{3/4}, B_N ~ N^{-1/4}, and the exact
conservation law c A_N + B_N = 1. That is the whole statement.