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-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.
-
-========================================================================
-1. THE MODEL AND THE RESULT
-========================================================================
-
-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.
-
-========================================================================
-2. POSITIVE FLUX AND FINITE STRIP TRANSFERS
-========================================================================
-
-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
+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.
- 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.
+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
-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
+ rho = 1/sqrt(2 + sqrt 2) = 1/(2 cos(pi/8)).
- 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).
+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.
-2.4 The splitters.
+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.
-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.
+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.
-2.5 The four local identities.
+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
-Write R_i for the action on slots i, i+1. Then
+ a(u) = cos(2u - 3 lambda),
+ L(u) = sin(2 lambda + u) sin(3 lambda + u) = (C + a(u))/2,
- 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,
+the weights are
-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).
+ 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).
-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:
+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.
- 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.
+These are Nienhuis's integrable O(n) weights at n = 0, the self-avoiding
+walk, normalised as in Glazman and Manolescu.
-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).
+https://doi.org/10.1016/0378-4371(90)90325-M
-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.
+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.
-2.7 Decay and rational continuation.
+https://arxiv.org/pdf/hep-th/9303161
-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.
+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
-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
+ 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_{M < L} q_H(M) <= 1/c.
+ rho^C_H * sum over M < L of 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.
+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.
-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
+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.
-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
+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 (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
+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},
-(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
+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 we may impose vacant ends or
-either source. For epsilon = 0, 1:
+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)},
+ 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)},
+ 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 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).
+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 now satisfies exchange and reduction:
+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.
+ 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.
+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 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
+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.
-
-Pi-shift: shifting u_k by pi multiplies every tile weight by the occupancy
+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. D_k fixes the empty
-coordinate, so uniqueness gives P_N(X + pi e_k) = D_k P_N(X).
+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.
-========================================================================
-3. THE POLYNOMIAL VACUUM
-========================================================================
+https://arxiv.org/pdf/math-ph/0410061
-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.
+https://arxiv.org/pdf/1411.7020
-Write X = (u_1,...,u_N) and a_i = a(u_i). Let (the Pfaffian form)
+https://arxiv.org/pdf/math-ph/0504032
- 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) ].
+Write X = (u_1, ..., u_N) and a_i = a(u_i). Define
-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.
+ D(a,b) = a^2 + b^2 + 2 d a b - d^2,
-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(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) ].
- p_N(Y, c) = p_{N-1}(Y) prod_{z in Y}(z + C). (flat)
+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}), (delete)
+ 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). (merge)
+ p(Y, f_{-2}, f_2) = p(Y, f_0)(f_{-2} + f_2 + rho)
+ prod_{z in Y}(z + f_0).
-Here the subscript of p is its number of arguments. Both formulas also hold
+The subscript of p is its number of arguments, and both formulas also hold
with all shifts negated.
-Proof.
-Degree. After multiplying by all D-factors, the Pfaffian becomes an
+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 gives degree at most
-N-1 in each variable.
+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 (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.)
+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) 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.
+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 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.
+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 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
+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 such a pair as g_0, g_6 with g_j = cos(v + j lambda); then
+Write that 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})
+ 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 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.
+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.
-Now wrap the polynomial around the vacuum vector: set W_N = p_N P_N.
+Wrap the polynomial around the vacuum vector by setting 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.
+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.
+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).
+ 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
+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_{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)
+ 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 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.
+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_{j<N, j != i} (b - beta_j)/(beta_i - beta_j).
+ 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
-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_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).
+ 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 exactly p_N.
+prescribed node values. Scalar interpolation shows the empty coordinate
+of W is 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
+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(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).
+ 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
+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.
- (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.
+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; the rest follows by continuation.
+dividing sines, and the remaining cases follow by continuation.
Stationarity. Form the polynomial residual
- V = (prod_{j=1}^N L(u_j)) (T_N - I) W.
+ V = ( prod over j = 1 to N of L(u_j) ) (T_N - I) W.
-At each last-variable node it vanishes, because the node data came from the
+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_{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
+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
+
+ 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
@@ -645,179 +594,175 @@ 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).
+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_{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.
+ 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; 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
+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,
+ 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
+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). (gluing)
+ 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. 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.
+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
-4.2 The arch formula.
+https://arxiv.org/pdf/1510.02721
-Let eta = 2d - 1 = sqrt(2) - 1 (so that c = eta C and C = eta(2C + c)) and
+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).
+ 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).
+ m(X) = sum over i = 1 to N of a_i - eta [z^(N-1)] g_X(z),
-The right side is read by continuation at removable singularities.
+the right side understood 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.
+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.
-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
+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^-,
+ (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.
+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 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
+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 simply be omitted.
+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
+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.
+ (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
+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).
+ (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.
+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. Call the right side of the arch formula m~. The scalar
+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_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).
+ 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
+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.
+ m(u) = A(u)U(u)/(1 - B(u))^2 = (a(u) - c)/(1 + d)
+ = a(u) - eta(a(u) + rho).
-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
+In the size-zero quotient convention use g_0 = 1 and [z^(-1)] g_0 = 0. This
+completes the induction. End of proof.
- m_N = N C - eta * lim_{a_i -> C} [z^{N-1}] g_X(z).
+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
-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.
+ m_N = N C - eta lim_{a_i -> C} [z^(N-1)] g_X(z),
-4.4 Increments versus the bridge mass.
+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.
-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.
+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
+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.
@@ -825,90 +770,81 @@ 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
+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_{j<i} Kbar(i,j) - sum_{j>i} Kbar(i,j) )
+ 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 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.
+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 identity over -L <= i <= L. Reversal pairs and cancels all internal
-arches; the uncancelled terms cross the two ends of the interval. After
+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 transfer decay, and paths
-leaving an arbitrarily large lateral window are controlled for each fixed
-pair of ports. Dominated convergence gives the exact identity
+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_i Z(p,i).
+ 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_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.
+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; 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
+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_i Z(p,i) <= (A/(1-B)) sum_{r>=0} ((1/(2c)) 2cB)^r B_N
- = (A/(1-B)^2) B_N.
+ 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 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.
+Combine with the exact identity. End of proof.
-For generic X write b = cos(w) and
+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_{i=1}^N
- (b - a_i)(cos(w - 2 lambda) - a_i)(cos(w + 2 lambda) - a_i),
+ 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.
+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
+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), 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
+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)
@@ -919,612 +855,549 @@ 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
+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.
-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.
+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
++ 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).
+ 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.
+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. (homogeneous)
-
-5.2 Nonsingularity and the integral representation.
+ Q(b) = (b - C)^(2N+1) (b - c)^N g(b)
+ = P(h) + b J(h) + b^2 E(h), deg g = N.
-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
+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
+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)
+ 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_{i=1}^N (y_i - x)/(y_i + x) )
- omega(x)/x dx, (integral)
-
-where E is the expectation over the positive law.
+ m_N = (eta/pi) int_0^1 ( 1 - E prod over i of (y_i - x)/(y_i + x) )
+ omega(x)/x dx,
-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.
+the expectation taken over the positive law.
-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.
+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.
-Put z = (8h - 1)^{-1} to send them to infinity, with inverse relation
+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), 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
+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 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 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).
+ 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.
+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 upper and lower boundary values across the cuts,
+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).
-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
+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 comes from subtracting the bounded-at-infinity representation at zero;
-its integrand is integrable there because Et(-y)/y is a polynomial.
+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)).
-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. 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,
-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,
- 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
-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),
- 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.
-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.)
+https://arxiv.org/pdf/0904.2602
-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.
+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.
-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. 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
-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.
- [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:
+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. (moment-D)
+ = (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 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
+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 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
+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_{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).
+ 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. 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).
+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
-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
+ 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).
- sum_{r=0}^N z_r^k / prod_{j != r}(z_j - z_r) = 0, 0 <= k < N.
+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
-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.
+ sum over r of z_r^k / prod over j != r of (z_j - z_r) = 0, 0 <= k < N.
-========================================================================
-6. THE SMALLEST COORDINATE AND THE STRIP EXPONENT
-========================================================================
+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 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.
+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.
-6.1 Reduction to inverse moments.
+The integral formula gives at once
-The integral formula immediately gives
+ E y_1^(-1/4) ~< m_N ~< E ( sum over i of y_i^(-1) )^(1/4).
- E y_1^{-1/4} ~< m_N ~< E ( sum_{i=1}^N y_i^{-1} )^{1/4}. (inverse)
+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
-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} }.
+ 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. 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).
+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};
-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.
+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.
-For comparison use the generalized Bures--Laguerre family, in ordered
-coordinates and with parameter a > 0, having density
+The reference law is the generalised Bures-Laguerre family, in ordered
+coordinates with parameter a > 0, of 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.
+ (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 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
+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!.
- Cov(F(T), G(T)) = (1/2) E[ (F(T) - F(T'))(G(T) - G(T')) ] >= 0,
+https://arxiv.org/pdf/1410.6883
-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.
+Association and monotone comparison. Let I be an interval in (0, infinity),
+n >= 1, and w_1, ..., w_n positive smooth on I. If
-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.
+ 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,
-LEMMA 6.2 (Laguerre normalization and coordinate bounds).
-For fixed a > 0 and every n >= 1,
+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.
- Z_n(a) = prod_{i=0}^{n-1} Gamma(a + i) prod_{0<=i<j<n} (j - i)/(2a + i + j).
+Proof. The mixed derivative of each pair is
-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,
+ d^2/(dx dy) log K(x, y) = 2/(y - x)^2 + 1/(x + y)^2 > 0.
- Prob_{n,a}(r_k < h) <= min{ 1, ( C_a h n^2 / k^3 )^{D_k} }, h > 0.
-
-In particular
+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.
- 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)
+The law of total covariance then splits into two nonnegative terms:
+conditional covariance by induction, and covariance of the two conditional
+expectations, nonnegative by
-with absolute constants in the last three bounds.
+ Cov(F(T), G(T)) = (1/2) E[ (F(T) - F(T'))(G(T) - G(T')) ] >= 0,
-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
+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.
- Pf[ (x_j - x_i)/(x_j + x_i) ] = prod_{i<j} (x_j - x_i)/(x_j + x_i).
+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.
-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.
+https://doi.org/10.1051/ps/2013049
-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
+https://doi.org/10.1214/16-AOS1478
- 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).
+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.
-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).
+Laguerre normalization and coordinate bounds. For fixed a > 0 and every
+n >= 1,
-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
+ Z_n(a) = prod over i < n of Gamma(a + i)
+ prod over 0 <= i < j < n of (j - i)/(2a + i + j).
- 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
+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,
- 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).
+ Prob_{n,a}(r_k < h) <= min{ 1, ( C_a h n^2/k^3 )^(D_k) }, h > 0.
-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
+In particular
- (1/Gamma(a+n-1)) int_{Kn}^inf s^{a+n-2} e^{-s} ds <= e^{-Kn/2} 2^{a+n-1}.
+ 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),
-Choosing K large makes the tail uniformly (exponentially) small.
+with absolute constants in the last three bounds.
-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
+Proof. Normalization. Write the pair product as a Vandermonde determinant
+times the Schur Pfaffian with entries (s - r)/(s + r); the identity
- Prob_{n,a}(r_k < h)
- <= h^{D_k}/(a^k k!) * Z_{n-k}(a+k)/Z_n(a) (Z_0 = 1).
+ Pf[ (x_j - x_i)/(x_j + x_i) ] = prod over i < j of (x_j - x_i)/(x_j + x_i)
-From Z_n(a),
+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.
- 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)).
+https://arxiv.org/pdf/math/0411280
-For gaps j - i <= k use 2a + 2i <= C_a k. For each i < k,
+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.
- sum_{j=1}^k log(1 + C_a k/j)
- <= k log(1 + C_a) + k log k - log(k!) = O_a(k).
+https://pure.tue.nl/ws/files/1920642/597510.pdf
-So the short gaps contribute O_a(k^2) = O_a(D_k). For larger gaps,
-log(1 + x) <= x gives at most
+https://arxiv.org/pdf/math/9905083
- sum_{i<k} (2a + 2i) sum_{j=k+1}^{n-1-i} 1/j <= 2 D_k log(n/k)
+The double moment with indices i, j is
-(including k = n, when the latter sums are empty). Furthermore
+ 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),
- prod_{i=0}^{k-1} Gamma(a+i) >= k^{D_k} e^{-O_a(D_k)}.
+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.
-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.
+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
-For any 0 < p < D_k, integration of that tail estimate gives
+ Prob_{n,a}(r_n > K n)
+ <= (Z_{n-1}(a)/Z_n(a)) int_{Kn}^{inf} s^(a+n-2) e^(-s) ds,
- E_{n,a} r_k^{-p} <= (D_k/(D_k - p)) ( C_a n^2/k^3 )^p.
+and the explicit normaliser gives
-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).
+ 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),
-The minimum at a = 1. The density of the smallest coordinate at s > 0 is
+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
- 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) ].
+ (1/Gamma(a+n-1)) int_{Kn}^{inf} s^(a+n-2) e^(-s) ds
+ <= e^(-Kn/2) 2^(a+n-1),
-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
+so a large K makes the tail exponentially small in n.
- prod_i q_s(r_i) >= 1 - 3s sum_i r_i^{-1}.
+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
-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.
+ Prob_{n,a}(r_k < h)
+ <= h^(D_k)/(a^k k!) Z_{n-k}(a+k)/Z_n(a), Z_0 = 1.
-6.2 Completing the theorem.
+The normaliser formula gives
-Compare the positive law with the reference laws to estimate both sides of
-(inverse).
+ 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)).
-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
+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
- d/dy log(omega(y) y^{1/2})
- = (1 - 2y - sqrt(2y/(1+y)))/(4y(1 - y)) >= -4, 0 < y <= 1/2.
+ prod over i < k of Gamma(a+i) >= k^(D_k) e^(-O_a(D_k)),
-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
+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.
- 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}.
+For 0 < p < D_k, integrating the tail gives
-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
+ E_{n,a} r_k^(-p) <= (D_k/(D_k - p)) ( C_a n^2/k^3 )^p.
- 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)) ).
+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.
-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
+The minimum at a = 1. The density of the smallest coordinate at s > 0 is
- omega(2z/(1+z)) (1+z)^{-(N+1)} e^{bNz}
+ 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 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,
+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
- E y_1^{-1/4} >= 2^{-1/4} (bN)^{1/4} E_{N,1} r_1^{-1/4} ~> N^{3/4}.
+ prod over i of q_s(r_i) >= 1 - 3s sum over i of r_i^(-1).
-Equation (inverse) now gives m_N ~ N^{3/4}.
+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.
-Extracting the bridge estimate. By monotonicity (Lemma 4.2) sum the lower
-increment bound over 1 <= j < N:
+Completion. Compare the positive law with the reference laws on both sides
+of the inverse comparison.
- (N - 1) B_N ~< m_N, so B_N ~< N^{-1/4}
+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
-(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
+ d/dy log(omega(y) y^(1/2))
+ = (1 - 2y - sqrt(2y/(1+y)))/(4y(1 - y)) >= -4, 0 < y <= 1/2,
- 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
+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
- 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
+ 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).
-Substitution uses the product-to-sum identities
+Lower estimate. Set y_i = 2 z_i/(1 + z_i), with 0 < z_1 < ... < z_N < 1. The
+transformed pair kernel is
- 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.
+ 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)) ),
-The last two follow from the preceding formulas for s_1 s_5 and s_6 s_2.
-Substituting verifies every table entry.
+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
-A.3 Unitarity R(x) R(-x) = I.
+ 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
-The one-strand sector reduces to multiplication of two matrices with
-diagonal entries A and off-diagonal entries B; the even sector requires
+together with the product-to-sum identities
- 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.
+ 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.
-A cap composed with a cup creates a loop and so vanishes. Each identity
-follows from the weights A.2.
+The last two follow from the preceding s_1 s_5 and s_6 s_2, and
+substitution verifies every entry.
-A.4 The braid identity.
+Unitarity. The one-strand sector is multiplication of two matrices with
+diagonal entries A and off-diagonal entries B, and the even sector needs
-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
+ 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,
- (I (x) S_s) Rhat_s(x) (S_s^T (x) I).
+each following from the weights, a cap composed with a cup contributing a
+loop and so vanishing.
-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.
+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; 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
+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; the scalar denominators are
-pi-periodic. Hence
+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}. 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}.
+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.
-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.
+This completes the proof of the four local identities, hence of the strip
+power with uniform constants.