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CRITICAL HONEYCOMB BRIDGES: A COMPLETE PROOF OF A LOGARITHMIC LENGTH WINDOW
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Version 2. Rewritten for clarity, brevity, and approachability.
Target reader: a bright high-school student who knows linear algebra,
basic complex numbers, and how to read a summation. Everything else --
the honeycomb lattice, self-avoiding walks, transfer matrices,
Pfaffians, vertex operators -- is introduced from scratch here.
What is rigorous here, and what is streamlined. Read this before
proceeding.
* Chapters 0-4 (motivation, setup, the finite cylinder transfer
matrix, the three exact identities, and the vertex-operator
reformulation of the colour sum): full proofs at the source's
level. The local Yang-Baxter and cap/triangle cell algebra is
reduced to a small closed computation on sine weights, whose
table is cited from Section 01.tex.
* Chapters 5-10 (long-circle expansion, positive norms, winding
pressure, strip crossing, two-port bound, enclosing-loop, and
half-plane growth): theorem statements and the main STRUCTURAL
steps of each proof are complete. Several heavy analytic lemmas
(typically the fine numerical estimates in the root-continuation
argument of Ch 7, the Hermite-remainder computation of Ch 8,
and the length-bin accounting of Ch 9) are condensed into
"proof outline" form with an explicit pointer to the source
section for the exhaustive detail. Two key asymptotics
(J_n ~ sqrt3/(6n) from Ch 6, and B_H ~ c H^{-1/4} from Ch 8)
are however stated and their argument structure is set out in
the file.
* Chapter 11 (the logarithmic window theorem): full proof, using
only the results of Chapters 5-10 as black boxes.
The companion source paper
("Cylinder-amplitudes-and-logarithmic-bridge-length-windows-on-the-
honeycomb-lattice") contains verbatim proofs of each lemma used as a
black box here. Where this document says "see Section XX.tex", that
means "the full verbatim proof is in the source, at the indicated
section"; the reader wanting every inequality is directed there.
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CONTENTS
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Abstract
Chapter 0. The story in one page
Chapter 1. Honeycomb SAW and the critical weight
Chapter 2. The main theorem: a logarithmic length window
Chapter 3. Cylinder transfer matrix and local rules
Chapter 4. Colour sum by vertex operators
Chapter 5. Long-circle expansion
Chapter 6. Positive norms, nonzero amplitudes, J_n
Chapter 7. Winding pressure and cylinder height tails
Chapter 8. Strip crossing: B_H of size H^(-1/4)
Chapter 9. Enclosing loops to the polygon second moment
Chapter 10. Positive bridges, caps, and growth
Chapter 11. From moments to a logarithmic length window
Appendix A. Notation
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ABSTRACT
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Consider a critical self-avoiding walk on the honeycomb lattice. The
physical prediction (Nienhuis 1982; Lawler-Schramm-Werner 2004) is that
a critical crossing of a strip of height h has length of order h^(4/3).
Duminil-Copin-Smirnov (2012) proved the critical fugacity exactly on the
honeycomb. Krachun-Panagiotis (2026) proved a polynomial upper bound for
the total crossing mass and quantitative sub-ballisticity of uniform
walks.
"of order h^(4/3)" is not a theorem, because a tiny subset of paths can
carry an arbitrarily large share of any fixed sum. This paper closes
that loophole. We prove that length-h^(4/3) crossings carry a definite
share of the total mass: they are not a small, isolated collection.
THEOREM (logarithmic window; sharp in power, up to logs).
There are constants c, C > 0 and h_0 such that for every h >= h_0,
with
R(h) = h (log h)^(1/64),
len_min(h) = h^(4/3) (log h)^(-1/8),
len_max(h) = 2 h^(4/3) (log h)^(1/2),
the total critical mass of bridges of integer heights H satisfying
1 <= H <= 2 R(h) and of length between len_min(h) and len_max(h) is
>= c h^(3/4) (log h)^(-C).
No subsequence, no parity condition, no single exact height. Just an
averaged statement with only logarithmic losses.
The proof has three parts, each interesting on its own:
(a) An exact cylinder computation via the integrable Yang-Baxter
loop model of the honeycomb with critical weights. This gives
exact finite expressions ("vacua") in terms of two symmetric
polynomials Q and R (a Pfaffian and a determinant), and their
asymptotics for large cylinder circumference.
(b) A positive second-moment bound on plane polygons of scale R.
This is the analytic price of "no long tail dominates".
(c) A "sewing" construction: a long arch plus two disjoint exterior
connectors become a strictly long bridge, and the input arch is
recoverable from the output bridge.
Part (a) is Chapters 3-9. Part (b) is Chapter 9. Part (c) is Chapter 11,
relying on the geometric lemmas of Chapter 10.
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CHAPTER 0. THE STORY IN ONE PAGE
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A path on a graph is a sequence of vertices v_0, v_1, ..., v_n where
v_{i+1} is adjacent to v_i. It is self-avoiding (the walk is a SAW)
if no vertex is visited twice. Write |omega| = n for the number of
vertices (not edges) that the path visits.
The honeycomb lattice is the planar graph with hexagonal faces and
three edges per vertex. Its self-avoiding walks are the canonical
combinatorial model for a polymer; in statistical physics they are the
"n -> 0" limit of the O(n) spin model.
Criticality. There is a unique number X in (0,1) -- the critical
fugacity, or inverse connective constant -- such that the number of
length-n SAWs is roughly X^(-n) times a sub-exponential factor. For
honeycomb,
X = 1 / sqrt( 2 + sqrt(2) ).
This is Duminil-Copin-Smirnov's theorem (Annals 2012).
The physics. Conformal invariance predicts an exponent 3/4 for SAW
the "distance/length" scaling in a strip; that is, a critical SAW that
crosses a strip of height h has length of order h^(4/3). Boundary
exponents for arches (SAW with both ends on the boundary) is 5/8.
What this paper proves. A quantitative window length ~ h^(4/3),
height ~ h, with total mass >= h^(3/4) (log h)^(-C). The proof is a
mixture of exact integrability and positive-probability geometry.
What it does NOT prove. The mean length of a critical crossing is
asymptotic to a constant times h^(4/3); that a specific height H is
attained; that any single endpoint is attained. Those are separate.
What it does prove.
- B_H = total mass of height-H bridges, = Theta(H^(-1/4)).
- A_g = total mass of boundary arches with endpoints at gap g,
= Theta(g^(-5/4)).
- All plane polygon second moments of scale R are O(R^(2/3) polylog).
- The logarithmic window of the abstract.
- Standard bridge-renormalisation facts (irreducible law, height
tail ~ t^(3/4), diameter tail ~ r^(-3/4)).
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CHAPTER 1. HONEYCOMB SAW AND THE CRITICAL WEIGHT
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1.1 The triangular lattice and its faces.
Take the triangular lattice in the plane: vertices Z e_0 + Z e_1 with
|e_0| = |e_1| = 1 and angle(e_0, e_1) = 60 degrees. The elementary
faces are unit equilateral triangles.
The honeycomb lattice is the graph whose vertices are these elementary
equilateral triangles; two triangles are adjacent if they share a side.
Equivalently: put a point at the center of each unit triangle and join
centers of triangles that share a side. The result is a hexagonal
tiling: every vertex has three edges, every face is a hexagonal loop
of six edges. The graph is the standard honeycomb (its vertices are
the "hexagon centers", i.e. the triangle centers).
Ports. The midpoint of a honeycomb edge is called a port. A honeycomb
edge is the segment joining two adjacent triangle centers, and it
crosses the shared side of the two triangles at exactly its midpoint.
Equivalently, a port is the midpoint of a triangle side shared with
one neighbour.
The choice of ports rather than vertices is a technical convention
but a very useful one:
- Ports are the natural "gluing points" for two paths. If two
port-paths both terminate at the port P, we may concatenate them
(delete the common port), and weights multiply.
- Ports sit on the shared side of the triangles; the set of ports
at each triangle centre is exactly {L, B, R, T} (left, bottom,
right, top); every cell has these four ports "in the same cyclic
order", so the honeycomb behaves like a rhombus grid.
1.2 Port-paths, self-avoidance, and critical weight.
A port-path is a sequence
P_0, v_1, P_1, v_2, P_2, ..., v_n, P_n
where each P_j is a port, each v_i is a triangle centre (a honeycomb
vertex), and P_{i-1} and P_i are two distinct ports of v_i, joined
in v_i by a single arc through the centre. In other words, a port
path enters via one port of v_i, exits by another, and moves on to
the adjacent centre across the exit port.
The length of the port-path, written |omega|, is the number of
triangle centres it visits. This is not the number of edges; each
centre visit corresponds to one "step". Our convention throughout:
length = number of centres visited.
Self-avoidance. A port-path is self-avoiding if it visits no triangle
centre twice; ports are automatically visited with degree <= 2 by
construction (each port has one incoming and one outgoing step at most,
except the two endpoints).
Critical weight. Define
X = 1 / sqrt( 2 + sqrt(2) ) ~ 0.541196.
The weight of the port-path is mu(omega) = X^|omega|. The choice of X
is essential; it is the unique value of the fugacity X at which the
self-avoiding walk is critical, in the sense that:
sum_{|omega| = n} X^n is bounded above and below
by constants (independent of n), times sub-exponential factors,
so that X is the reciprocal of the connective constant.
1.3 Theorem (Duminil-Copin-Smirnov, 2012).
The connective constant of the honeycomb lattice equals
c = sqrt( 2 + sqrt(2) ),
that is, the number of n-step honeycomb self-avoiding walks from a
fixed starting vertex grows like c^n times a sub-exponential function
of n. Equivalently, the critical fugacity is X = 1/c = 1/sqrt(2+sqrt2).
The proof uses a "parafermionic observable" -- a weighted alternating
sum of complex numbers over walks ending at points of a large box's
boundary -- whose local invariance around every small triangle turns
into an exact boundary flux identity. That identity determines the
only value of X at which this alternating sum can survive a scaling
limit; that value is 1/sqrt(2+sqrt2).
For us, X is a fixed number. We work at X = 1/sqrt(2+sqrt2) throughout.
1.4 Why X = 1/sqrt(2+sqrt2) in our port (centre-counted) algebra.
The classical form of the theorem counts edges of a honeycomb walk
started at a vertex. Our "port-paths" count centres and use a port
at each end. These two conventions are related by bounded shifts
of n and the start/end. At criticality both have the same base c:
the number of "port-paths of centre-length n" is c^n up to sub-exponential
factors, with c = sqrt(2+sqrt2). The corresponding weight factor is
X = 1/c. When a port-path is glued to another port-path at a shared
port, the two centre-lengths add and the weights multiply, so weights
are a multiplicative gluing rule. This is the fundamental fact behind
all finite transfer matrices in this text.
Notation for the rest of the paper.
For a port-path omega,
mu(omega) = X^|omega|, X = 1/sqrt(2+sqrt2),
|omega| = number of triangle centres visited.
A bridge of height H is a port-path from a fixed port on horizontal
grid line "y = 0" to any port on horizontal grid line "y = H",
with every visited centre strictly between the two lines. Let
B_H = sum over bridges of weight mu. Set B_0 = 1 (empty bridge).
An arch of gap g is a port-path in the upper half-plane, starting
at a fixed port on the boundary and ending at the boundary port at
signed horizontal distance g, and meeting the boundary again only
at its two ends. Let A_g = sum of weights of such arches.
Grid lines. "Horizontal grid line" means a line of parallel
triangle sides in one fixed lattice direction; consecutive lines
are at unit spacing. There are three such directions, rotated by
120 degrees; all three are equivalent for us.
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CHAPTER 2. THE MAIN THEOREM: A LOGARITHMIC LENGTH WINDOW
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2.1 The two theorems we actually prove.
The paper has one headline result, the window theorem, and one support
theorem that packages all finite estimates. Both are stated here so
the reader knows where the rest of the paper is going.
THEOREM A (logarithmic window).
There are constants c, C > 0 and h_0 > 0 such that for every
real h >= h_0, if
R(h) = h (log h)^(1/64),
len_min(h) = h^(4/3) (log h)^(-1/8),
len_max(h) = 2 h^(4/3) (log h)^(1/2),
then
sum_{1 <= H <= 2 R(h)} sum_{omega in B_H,
len_min(h) <= |omega| <= len_max(h)}
mu(omega) >= c h^(3/4) (log h)^(-C).
The inner sum is over bridges of height H whose length (number of
triangle centres) lies between len_min and len_max; the outer sum
is over all integer heights H in [1, 2 R(h)]. All sums are of the
critical weights mu(omega) = X^|omega| at X = 1/sqrt(2+sqrt2).
THEOREM B (finite boundary and polygon estimates).
There are positive constants, independent of all parameters, such that
(i) B_H = Theta( H^(-1/4) ) as H -> infinity.
(ii) A_g = Theta( g^(-5/4) ) as g -> infinity.
The lower bound of (ii) can be restricted to arches of diameter at
most C g, for a fixed sufficiently large C.
(iii) For every sufficiently large fixed T, bridges of height H whose
transverse displacement from the starting normal stays at most
T H have mass at least c_T H^(-1/4) and length-weighted mass at
most C_T H^(13/12). Either weak sign of terminal transverse
displacement may be specified in the mass lower bound.
(iv) For arches and bridges in a strip of height H, from a fixed wall
port, the mass of paths of diameter exceeding r >= C H is
<= C H^(-1/4) exp( -c r/H ).
(v) The polygon second moment: for all sufficiently large R,
sum_{R <= diam P < 2R} mu(P) |P|^2
<= C R^(2/3) (1 + log log R).
Here mu(P) = X^(number of visited centres of P), as for paths.
Here "f = Theta(g)" means there exist positive constants c, C
(independent of the size parameter) with c g <= f <= C g. In (v)
polygons are counted modulo primitive lattice translations.
2.2 What is hard about Theorem A.
The physics prediction is that the mean length of a critical crossing
of height h is asymptotic to a constant times h^(4/3). One way to try
to prove this would be: show B_H ~ H^(-1/4), and compute the first
moment of |omega| in B_H, and find it is h^{4/3}.
The obstruction is that a first moment is an average; it does not say
that any positive fraction of the paths have length near h^{4/3}.
Imagine a single path of length 10^100 carrying half of the mass of
B_H and all other paths of length <= h^{4/3}: the first moment would
still be huge, but the mass at length scale h^{4/3} could be negligible.
Theorem A rules this out: it says that lengths in a window
of width (log h)^(5/8) around h^(4/3), summed over an interval of
heights of width h (log h)^(1/64) around h, carry mass at least
h^{3/4} (log h)^(-C). It does not pinpoint a single height or a
single length, but it excludes any scenario where the mass "leaks
out" to huge lengths or to a single pathological configuration.
The "logarithmic" in the title is because the ratio
len_max(h) / len_min(h) = h^0 * (log h)^{5/8}: a polylog factor,
not a power. Up to those logs, the length scale is exactly h^{4/3}.
2.3 The three ingredients of the proof.
(a) Exact integrable normalisation on a cylinder.
When the honeycomb is wrapped on a cylinder of circumference n,
its transfer matrix factorizes into local "rhombus" weights that
satisfy the Yang-Baxter relations. This turns certain sums over
loop configurations into finite closed expressions involving
Pfaffians and determinants. Chapters 3-8 construct those
expressions and evaluate their n -> infinity asymptotics.
The most important concrete outputs are:
J_n ~ (sqrt3)/(6 n) (winding-loop mass),
B_H ~ H^(-1/4) (crossing mass in a strip).
The first identity says "a critical cylinder of circumference n
contains a winding loop with probability roughly 1/n", and its
precise constant sqrt(3)/6 is essential for the pressure argument.
(b) Positive second-moment bound for plane polygons.
Chapter 9 proves
sum_{R <= diam P < 2R} mu(P) |P|^2
<= C R^(2/3) (log log R).
This is a bound on the "mass-weighted second moment of the polygon
length", an exact analogue of a variance bound. It is what blocks
a single very long polygon from carrying all the mass.
(c) Positive geometric sewing.
Chapters 10-11 build the following machine. Start with an arch
(an open path with both ends on the boundary). Attach two disjoint
"exterior connectors" that go off to opposite side walls, forming
a strict bridge of prescribed height and bounded diameter. Count
how many input configurations map to the same output bridge, and
show that the input is recoverable up to bounded (or polylog)
multiplicity. Theorem A follows from combining the resulting
mass balance with the moment bounds of (a) and (b).
2.4 Plan of the paper.
Chapter 3. Cylinder transfer matrix and its local rules.
Chapter 4. Colour sum via vertex operators.
Chapter 5. Long-circle expansion.
Chapter 6. Positive norms and J_n.
Chapter 7. Winding pressure and cylinder height tails.
Chapter 8. Strip crossing and B_H.
Chapter 9. Enclosing loops; two-port identity and second moment.
Chapter 10. Positive bridges, caps, and growth.
Chapter 11. From moments to a logarithmic length window.
Chapters 3-9 are analytic (they prove the amplitude statements); they
can be read mostly independently of Chapter 11. Chapters 10-11 are
purely combinatorial / geometric and use only the statements (not
the proofs) of Theorem B.
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CHAPTER 3A. CYLINDER TRANSFER MATRIX: DEFINITIONS AND LOCAL RULES
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3A.1 Parameters.
Define
q = exp( i pi/4 ), (a primitive 8th root of 1)
k = 2 + sqrt(2),
X = k^(-1/2) = 1/sqrt(2+sqrt(2)), (the critical weight),
kappa = k - 1 = 1 + sqrt(2).
All arithmetic is in the complex numbers; the weights below are all
expressible in sin and cos of pi/8. The choices of q and X are
chosen to be compatible with the parafermionic weight X; any half-power
of q is taken in the principal branch.
3A.2 The rhombus cell B(s).
We model the honeycomb locally as a square (rhombus) with four ports
in cyclic order L, B, R, T (left, bottom, right, top). When two
neighbouring triangles are joined across a shared side, the resulting
quadrilateral has this same structure. The number s is the ratio
"right rapidity / left rapidity", and encodes the local geometry: at
s = q the rhombus splits into two unit equilateral triangles, giving
the honeycomb at critical activity; at s = q^2 we get the reflected
picture (obtained by rotating by 90 degrees).
Write s = z_right / z_left. A "diagram in the cell" is a
non-crossing matching of a subset of the four ports {L, B, R, T}: it
is either empty, a single arc joining two ports, or two disjoint arcs.
Each rhombus cell carries weights depending on s:
eta = (3/8) * ( -i * log(s) ) , (so s = e^{i theta} with eta = 3 theta / 8)
D = sin(5 pi / 4 + eta) sin(5 pi / 8 - eta),
u = sin(5 pi / 4) sin(5 pi / 8 + eta) / D,
U = sin(5 pi / 4) sin(eta) / D,
v = - sin(5 pi / 8 + eta) sin(eta) / D,
w = sin(5 pi / 8 + eta) sin(5 pi / 4 - eta) / D,
W = - sin(15 pi / 8 + eta) sin(eta) / D.
Assignments:
- empty diagram: weight 1.
- one arc joining L to T, or B to R (opposite sides "topological pair 1"):
weight u.
- one arc joining L to B, or T to R (opposite sides "topological pair 2"):
weight U.
- one arc joining L to R, or B to T (opposite sides "vertical"):
weight v.
- two disjoint arcs, one L-T and one B-R:
weight w.
- two disjoint arcs, one L-B and one T-R:
weight W.
The label of the cell is B(s); it is a 6x6 matrix on the six local
configurations of a rhombus. Its matrix entries are the weights
1, u, U, v, w, W above in the appropriate positions; all other
configurations (same-side arcs, non-crossing only) are zero.
Remark (honeycomb point). At s = q, one has the interpretation:
split the rhombus into two unit equilateral triangles, one with sides
L and T, one with B and R; each triangle supports either a single arc
between two distinct side-midpoints via the center (weight X), or
nothing (weight 1). This yields a walk that goes through cell centers
with weight X per visited center. This is exactly our critical
honeycomb model from Chapter 1. At s = q^2 the reflected picture
holds. All our applications are at s ∈ {q, q^2}.
3A.3 The triangle map P and the cap map C.
The "triangle map" P is the local object that replaces one line by
two neighbouring lines, at neighbouring rapidities (q m, m/q). It
has three allowed states:
- empty: weight 1.
- arc between one input and one output
(three possible pairings): weight X.
- nothing else.
So P is a 1 x 3 matrix on the outputs.
The "cap map" C is the transposed version: it replaces zero lines by
two. It has two allowed states:
- empty: weight 1.
- a cup joining the two outputs: weight 1.
There are three basic identities that relate B(s), P, C. All hold
at the level of weights, by direct substitution in the sine formulas:
(Inversion) B(s) B(1/s) = 1, for all s.
(Triangle) B(q^2) = P P^T, (notation: P on one side,
transpose on the other).
(Cap) B(q^3) = C C^T.
The inversion identity says that a double cell in which the two lines
are swapped back is the identity in the cell algebra; the triangle
and cap identities express the "special" rapidities q^2 and q^3
in terms of the fundamental triangle P and the fundamental cap C.
Both are proved by direct substitution of the sine expression for
u, U, v, w, W and the definitions of P, C.
3A.4 Sliding lemmas.
A priori, a row of cells in a cylinder is a product
B(s_1) B(s_2) ... B(s_N) of cells acting on ports. Sliding lemmas
let us shuffle adjacent cells without changing the total weight:
(Ordinary braid)
B_{12}(s) B_{23}(s r) B_{12}(r) = B_{23}(r) B_{12}(s r) B_{23}(s).
(Triangle slide)
(P^T (x) 1) B_{23}(q s) B_{12}(s / q)
= B(s) (1 (x) P^T) (one orientation)
(Cap slide)
B_{23}(s) B_{12}(q^3 s) (1 (x) C)
= C (x) 1 . (one orientation)
(Quarter-turn)
R(s) = R(q^3 / s) up to an explicit "bend" factor described
below.
All the identities are local, in the sense that they hold at a cell
level before being assembled in a row. Once they hold locally,
they act simultaneously on all ports in a row. Proof: substitute
the sine expressions for u, U, v, w, W; both sides become equal
products of cosines of pi/8 with common factors. The original
source gives the exact scalar checks in a small table.
The "bending a leg" rule is a consequence: in the inversion identity,
bending a port commutes with sliding a cap several cells, with an
extra phase given by (q^3). Precisely:
B_{23}(s) B_{12}(q^3 s) (1 (x) C) = C (x) 1 .
This gives that when a cap is slid across a swap of rapidity q^3
(i.e. across a cell at the special angle q^3), it can be turned into
an ordinary cap on the other side. The triangle slide is the
analogue for the q^2 case.
3A.5 Loops and diagram composition.
Assemble the local algebra into row diagrams:
- Lay out N cells in a row, one per column, each a rhombus
B(s_i), s_i = z_i / t. The parameter t is the row auxiliary
line parameter (homogeneous everywhere along the row).
- A "row" is a diagram that is non-crossing inside the row.
- Two rows compose vertically (glue each row's outputs into the
next row's inputs); if any occupation is incompatible (one side
has the port occupied and the other vacant), the term is 0.
- If a closed loop is formed during composition, we drop the term
(multiply by 0). This implements self-avoidance: no cycles.
With this rule, "row addition" is a linear map T on pairing states.
The "vacuum" is the vector of half-cylinder sums; T acts linearly
and preserves it (in the sense of Chapter 4).
3A.6 The half-cylinder vacua V^σ.
Consider a half-cylinder: a periodic-cylinder stripped at one
horizontal cut, so it is a disk with open boundary at that cut. A
"half-cylinder diagram" is a set of disjoint arcs inside the half-
cylinder whose ends are ports on the boundary cut; no closed loops.
Define, for sign σ = +1 or σ = -1:
V^σ(Z; t) = map on pairing states p (non-crossing matchings of a
subset of the N boundary ports). For each state p, the entry
(V^σ)_p(Z; t) is the sum of weights over all half-cylinder
diagrams that end at p, times:
- σ for every port on the seam that carries the auxiliary
line (i.e. that is "crossed horizontally"), counted once
per crossing;
- no extra factor for any vertical-port crossing.
Normalisation: (V^σ)_∅(Z; t) = 1, i.e. the "empty" state has
value 1.
Two facts (proved in the original Section 01.tex and not reproved
here since we will only use the resulting formulas):
Convergence. For |s| in a neighbourhood of q or q^2, V^σ is
absolutely convergent. This uses the parafermionic boundary
bound: a single-arc half-cylinder walk from a boundary source
terminates at boundary ports with total weight at most
1/cos(3 pi / 8), uniformly in the depth of the half-cylinder.
Once single arcs converge, combinatorial accessibility of every
pairing state (working downward from the slice, adding fresh rows)
shows that the row-addition map on the non-empty block has
spectral radius < 1, hence transitivity and absolute convergence.
Uniqueness. V^σ is the unique solution of T V^σ = V^σ with
(V^σ)_∅ = 1, in the convergent regime.
We also record the "physical identification": at s = q (the honeycomb
point), V^+ is exactly the mass of half-cylinder loop gas with
positive seam twist, and at s = q^2 it is the reflected picture; in
general the substitution s → q^3 / s implements vertical reflection
of the cylinder and sends σ = +1 to σ = -1.
------------------------------------------------------------------------------
CHAPTER 3B. THE TWO SCALAR POLYNOMIALS Q AND R, AND THE THREE IDENTITIES
------------------------------------------------------------------------------
3B.1 Two auxiliary polynomials of primary importance.
We introduce two symmetric polynomials built from the multiset Z = (z_1,...,z_N)
of N rapidities. Everything in the exact cylinder computation is
written using these two objects; their n -> infinity asymptotics are
the analytic heart of Chapters 5-8.
Kernel and prefactor:
h(x, y) = x^2 + y^2 + sqrt(2) x y,
K(x, y) = (x^2 + y^2) * h(x, y),
d(x, t) = (x - q^5 t) * (x - q^6 t).
Vandermonde:
F(Z) = prod_{ i < j } h(z_i, z_j) / (z_i - z_j).
(For N = 0 or 1, F = 1.)
Pfaffian numerator:
Q(Z) = F(Z) * Pf_ext [ (x^2 - y^2)/h(x,y) ]_{x, y in Z}.
Here "Pf_ext" means: Pfaffian of the N x N skew matrix
[(x^2-y^2)/h(x,y)], for N even. For N odd, augment by a last
column of all 1's and last row of all (-1)'s (standard bordered
Pfaffian), so the size stays even. Q is a symmetric polynomial in
z_1,...,z_N of degree <= N-1 per variable; the denominators cancel
because entries at coincident x=y are antisymmetric under x <-> y.
Determinant numerator:
M(x, y) = k x y / h(x, y),
R(Z) = F(Z)^2 * det M(Z, Z).
Here M(Z, Z) = the N x N matrix (M(z_i, z_j))_{i,j=1,...,N}. The
determinant is that of a symmetric matrix; R is a symmetric polynomial
of degree <= 2(N-1) per variable.
Physical combination (the Pfaffian normalisation):
Y(Z) = kappa^N * Q(Z)^2 / R(Z).
Note Y is dimensionless. The three key values Y, Y^2, 1 appear as
special cases of the colour sum O_Z(r) at r = kappa, kappa^2, 1.
Wave functions (projection kernels):
p_Z^+(s) = s^{+1} - M(s, Z) * M(Z, Z)^{-1} * (z^{+1})_{z in Z},
p_Z^-(s) = s^{-1} - M(s, Z) * M(Z, Z)^{-1} * (z^{-1})_{z in Z}.
Here M(s,Z) = row vector (M(s, z_1), ..., M(s, z_N)); and (z^{±1})_{z in Z}
= column vector (z_j^{±1})_j. These are meromorphic in s with at
most simple poles at z_j-like values; they appear naturally as
projection residuals under the M-kernel.
3B.2 The colour sum O_Z(r).
Introduce two colours "+" and "-" with complex amplitudes b_+, b_-
satisfying b_+ b_- = k/2; and a "0" colour with amplitude b_0 = k.
The total r is
r = b_+ + b_- + k.
We will choose r ∈ {1, kappa^2, kappa, ...}; the important cases in
the theorem are
r = 1 => O_Z(1) = 1,
r = kappa => O_Z(kappa) = Y(Z),
r = kappa^2 => O_Z(kappa^2) = Y(Z)^2.
Concretely b_+ b_- = k/2 and b_+ + b_- = r - k, and consistency of
the two values of r against kappa requires b_+ + b_- to be real and
non-negative, which is how r is chosen. Two natural choices are
(phi_+, phi_-) = (-7 pi / 8, -pi / 8) => r = 1,
(phi_+, phi_-) = ( -pi / 8, -pi / 8) => r = kappa^2,
(phi_+, phi_-) = (-7 pi / 8, -7 pi / 8) => r = 1 again,
(phi_+, phi_-) = (-7 pi / 8, pi / 8) => r = kappa.
The precise phase convention is immaterial for the identities below;
only r enters.
Scalar functions and colour-pair factors (u = z_j/z_i is the ratio
of rapidities of two columns; for i < j, "u" is z_j / z_i):
S(u) = (u + 1)^2 / [ (u - q)(u - q^{-1}) ],
P_{++}(u) = 1,
P_{--}(u) = 1,
P_{00}(u) = S(u),
P_{+0}(u) = P_{0-}(u) = S(u/q),
P_{+-}(u) = S(u/q) S(u/q^2),
P_{ba}(u) = P_{ab}(1/u) (for the transposed entries).
The two-phase analytic structure: S(u) has simple poles exactly at
u = q, q^{-1}, and is symmetric under u -> 1/u. These are the
places where the fusion/colour-sum rules of the cylinder model
single out the "+" and "0" channels; see Section 3A.
Colour sum. With respect to the N columns, define
O_Z(r) = ( prod_{ i < j } K(z_i, z_j) / R(Z)^2 )
* sum_{ alpha in {+,0,-}^N }
( prod_i b_{alpha_i} ) *
( prod_{ i<j } P_{alpha_i alpha_j}(z_j / z_i) ).
Rational continuation. All denominators are cleared by
R(Z)^2 / prod K of the prefactor; on coincident parameters
z_i = z_j the sums continue to rational values that we take as
the definition. This is the "confluent" convention: always
continue from generic distinct z_i.
3B.3 The three finite cylinder identities (Prop 3.1).
Let Z = (z_1, ..., z_N) be the rapidity multiset of the N columns.
Recall from 3A.6 the vacua V^σ. Consider the N-column regular
periodic cylinder wrapped on a circle of circumference N (all columns
in a row).
Marked-loop mass E_+.
Place the mark at the auxiliary port L of column 1 (the seam),
and also consider the row auxiliary parameter t. E_+(Z; t) is
the mass of configurations consisting of exactly one loop and
containing the mark.
Winding-loop mass J.
J(Z; t) is the mass of single-loop configurations that wind
around the cylinder (i.e. cross the seam exactly once, counted
mod row translations).
Colour sum O_Z(r).
Defined in 3B.2.
PROPOSITION 3.1 (finite cylinder identities).
For any N >= 0, any rapidities Z, and any t (row parameter) at
which the identities are well-defined (in particular any complex
t avoiding the poles below), the following hold.
(3.1.1) E_+(Z; t)
= 1 + kappa ( Q(Z, t) * Q(Z, q^3 t)
/ [ Q(Z)^2 * prod_{ z in Z } d(z, t) ]
- 1 ).
Here Q(Z, t) means: augment the multiset Z by the additional
rapidity t (so N+1 variables in total).
(3.1.2) Y(Z) * J(Z; t)
= (-1)^{N+1} / sqrt(2)
* [ p_Z^+(t) p_Z^-(q^3 t) + p_Z^-(t) p_Z^+(q^3 t) ].
(3.1.3) O_Z(1) = 1,
O_Z(kappa) = Y(Z),
O_Z(kappa^2) = Y(Z)^2.
All identities hold first at the physical parameters (s = q or s = q^2),
then extend to all complex parameters by rational continuation.
Special-branch consistency: replacing q by q^{-1} in the colour
factors has the same effect as exchanging + and - including
b_+, b_-.
Mark evaluation (subsidiary to 3.1.1): for any x ∈ Z,
E_+(Z; x) = E_+(Z; x / q^3), J(Z; x) = -J(Z; x / q^3),
with the common first value of E_+ given explicitly by the
occupied-port slice-gluing formula, and with positivity identities
used in Chapter 6. These also hold for any x ∈ Z, and in particular
at x = z_1 they give the mass of a single polygon through a top
port at the first column in the honeycomb evaluation.
3B.4 Structure of the proof of Proposition 3.1.
The proof of Prop 3.1 has three steps, carried out in the original
Section 01.tex. We record the outline, since each step is
independently useful.
Step 1. Diagram identities.
Treat the rhombus B(s) as a two-line swap. Its four ports
L, B, R, T are (old first, old second, new second, new first); the
argument s is right / left. Verify:
- Inversion (B_{12}(s) B_{12}(1/s) = 1),
- ordinary braid relation (the local Yang-Baxter equation),
- triangle slide (any P can be moved across a swap by sliding),
- cap slide (any C can be moved similarly).
All these scalar checks reduce to the sine expression for
u, U, v, w, W; explicit small tables appear in the source.
Consequences.
- Row addition intertwines with adjacent rapidity swaps.
- Carrying the seam past a column multiplies the twist by a
sign σ per occupied port at that column; the sign is +1 for
even occupation parity of a cell (each cell has 4 occupied-end
slots, so parity is preserved).
Step 2. Polynomial denominators Q and R.
Show that under the fundamental reductions of the vacuum pairing
coefficients (which are the operational meaning of the triangle and
cap slides), Q and R have the following behaviours as symmetric
polynomials:
Reduction values:
(i) pole-pair [l, q^3 l, Z] / (Z):
Q: (1 + q^3) l * prod_{z in Z} d(z, l)
R: kappa^2 * [(1 + q^3) l * prod_{z in Z} d(z, l)]^2
(ii) fusion [q m, m/q, Z] / (m, Z):
Q: sqrt(2) * m * prod_{z in Z} (z + m)
R: k sqrt(2) * m^2 * prod_{z in Z} (z + m)^2
(iii) removal at 0: Q: prod z, R: prod z^2.
(iv) removal at infinity: leading coefficient = smaller Q, R.
The verification of (ii) uses the Schur-complement kernel identity:
after pivoting on the pair (b,c) = (q m, m/q), with
f_m(s) = (m^4 + s^4)/(m^4 - s^4) and A(s,t) = (s^2 - t^2)/h(s,t),
A(s,t) + [A(s,c) A(t,b) - A(s,b) A(t,c)] / A(b,c)
= f_m(s) f_m(t) [ A(s,t) - A(s,m) + A(t,m) ].
For M the same identity holds with M replacing A and an additional
factor m^2, since M(s,t) = k s t / h(s,t) factors through A times
k s t / (s^2 - t^2) up to an m-dependent correction.
Consequently Q and R are the *unique* symmetric polynomials of the
stated degrees realising these reductions. This uniqueness
identifies the (almost-best) yellow vacuum states as specific
polynomial quotients of Q and R:
(V^σ_N)_p(Z; t)
= prod_j z_j^{o_j/2} / A_δ(Z) * psi_p(Z),
δ = 1 for σ = +1, δ = 2 for σ = -1,
where A_δ = Q if δ = 1, A_δ = R if δ = 2, and o_j ∈ {0,1} is the
occupation of port j in the pattern p. Each psi_p is a symmetric
polynomial in Z of degree <= δ(N-1) - o_j, independent of t.
Step 3. Gluing rules.
Glue two half-vacua (V^σ below and its reflection above) as discussed
in 3A.6. The upper half is obtained by reflection s → q^3 / s,
i.e. by replacing each column's rapidity by z_j^{-1} and the row
parameter by 1/(q^3 t). The upper vacuum has the same A_δ up to
monomials and is again t-independent.
For E_+: use σ = +1 above and below, and require both open ends at
the mark occupied and connected; this gives (3.1.1).
For J: use σ = -1 below and +1 above, and count winding loops by
the sign convention of the seam; this gives (3.1.2).
For O_Z(r): compute the residues at the coincident points q^2 and
q^3 (fusion and pole-pair reduction) and verify that the residues
reproduce the Euler product of the scalar factor S at each site; this
fuses the sum over colours site-by-site.
The three cases share a common structural fact: after multiplication
by A_δ(Z) A_{δ'}(Z) * prod d(z, t), each side becomes polynomial in
all parameters of the stated degree. Matching on all pair reductions,
on the two removals (0 and infinity), and on the pair-triple checks
(reductions at q^2 and q^3 combined) gives the identity by induction
in N, starting from the explicit formulas at N = 1, 2. The original
Section 01.tex carries out each induction step in full.
3B.5 What we will need downstream.
For the rest of the paper, the reader only needs the *statements*
(3.1.1)-(3.1.3), the fact that Q and R are Pfaffian/determinant
polynomials with the reductions (i)-(iii), and the following
consequences:
(i) Y(Z) is real and positive at positive real z_j (Fourier
positivity of the M kernel; Chapter 6).
(ii) J_n = J(1,...,1; t) is real and positive at the physical
parameters, and its large-n behaviour is dominated by the
leading term of Q and R (Chapter 6).
(iii) The long-circle expansion of Q(Z)^2 and R(Z)^2 (Chapter 5)
gives polynomial-in-log corrections to the Q/R leading terms.
------------------------------------------------------------------------------
CHAPTER 4. COLOUR SUM BY VERTEX OPERATORS
------------------------------------------------------------------------------
4.1 What this chapter is for.
The colour sum O_Z(r) from (3.1.3) is a finite sum over N "colour
words" alpha in {+,0,-}^N. It evaluates to Y(Z), Y(Z)^2, or 1 for
r = kappa, kappa^2, or 1. To extract the n -> infinity behaviour of
these values (needed to get J_n, B_H, and the positivity results), we
need a representation of O_Z(r) that is (i) exact at finite N and
(ii) analytically tractable at large N. This representation is
provided by vertex operators.
The route is:
1. Encode the N colour-word sum as a trace over a Fock space.
2. Extract the r = kappa, kappa^2, 1 cases as a coefficient [p^0]
of this trace after a Cauchy average.
3. Continuate the trace analytically in the circumference L = 8N/pi^3
etc.
This chapter writes down the trace, the coefficient extraction, and the
statement of the "scalar change of channel" that lets us compute the
trace as a Gaussian integral. Chapters 5-6 then analyse this integral
as N -> infinity.
4.2 The charge lattice Lambda.
Let e_0, ..., e_7 denote the eight symbols; work in the vector space
R^8 with one relation
D_0 + D_1 + ... + D_7 = 0,
and basis vectors D_0, ..., D_7 in the quotient. Let O denote the
cyclic shift O D_j = D_{j+1} (indices mod 8). Fix a symmetric
bilinear form by
D_0 . D_j = (2, -1, 1, -2, 2, -2, 1, -1)_j.
Conjugation with O determines the rest. On the quotient this form
is nondegenerate (checkable by cyclic Fourier transform) but is
indefinite; both signs occur.
Fix a distinguished vector a by
(1 - O + O^2) a = D_0.
Its row against the D_j is (a . D_j) = (1, 0, 0, -2, 0, 0, 1, 0).
Let Lambda = sum Z D_j be the lattice generated by the D_j.
Define three special vectors that will label the three colours:
a colour "+" : a,
a colour "0" : a - D_0 = (O - O^2) a,
a colour "-" : a - D_0 - D_1 = -O^3 a.
All three have square length a^2 = 8/3, and they are obtained from
each other by application of (a - sum of D_j's). Physically, these
are the "shifts" that move us between the three colours.
Two more pieces of bookkeeping.
delta = D_7, R_delta y = y - delta (delta . y) (reflection in delta),
M_b = O R_delta, h = -O S delta = -S D_0 S = (1 - O)^{-1},
H_b = h . h = 9/8.
Adjectives checked by multiplication of matrices:
S + S^T = 1, (metric self-adjoint)
S Lambda . Lambda subset Z, (integrality)
M_b h = h, delta . h = 1, a . h = 0.
The characteristic polynomial of M_b on the quotient is
det(1 - y M_b) = (1 - y^6)(1 - y^3)/(1 - y^2).
This is what appears in the trace formula below.
4.3 Vertex operators.
Fix a "period" parameter L > 0 and complex "torus coordinate" s = e^{2 pi i u / L}.
We need momentum-space and oscillator modes. Let |P> denote a "momentum
eigenstate" indexed by P in a copy of the complexified lattice; the
oscillators l_m(y) (m > 0, y in the lattice span) satisfy
[ l_n(y), l_m(z) ] = n * m * <y, z> * delta_{m + n, 0},
with positive modes l_m (m>0) killing constants, and l_0(y) acting as
y . P on |P>. Total degree D = oscillator degree + P^2 / 2.
Vertex operator:
V_beta(s) = exp( sum_{m>0} l_{-m}(beta) s^m / m )
* exp( - sum_{m>0} l_m(beta) s^{-m} / m )
* T_beta * e^{pi i S beta . P} * s^{beta . P + beta^2/2},
where T_beta is the momentum translation T_beta |P> = |P + beta>,
and S is the operator introduced above. In the vertex-operator
literature this is the standard construction; our only adaptation is
the use of S V (instead of the more common identity) in the cocycle.
Frenkel's Section 2.1 reference contains the precise statement in the
usual case; the present version with (S, T_beta) is a minor
generalisation that we verify by the "backward + forward" contraction
formula below.
Root modes and their relations: put e_b = -[V_b]_0 and f_b = [V_{-b}]_0
(where [ ]_0 means coefficient of s^0, i.e. the residue). These are
the standard "raise" and "lower" nilpotent operators at zero momentum.
They obey the SL_2-like commutation relation
[ f_b, e_b ] = l_0(b),
when b^2 = 2 (the "root" condition). For non-root b these relations
are those of a Heisenberg algebra with translation.
4.4 The trace.
For our application, the colour sum becomes the trace of a diagonal
operator on a space where:
- each of the N columns contributes one V_a-type field;
- a global conjugation inserts a twist (see 4.6 below).
Define the "trace" (formally a coefficient trace):
Tr( q^{D} * e^{-2 pi i D} * O_twist * prod_{i=1}^N V_a(s_i) ).
Here q = e^{pi i / 4} is the same 8th root of 1; D is the total
degree operator; e^{-2 pi i D} is the 8th-root-twist phase fixing q;
O_twist is a lift of O (the cyclic shift) acting on both momenta and
oscillators. The product of V_a(s_i) is complex ordering: the s_i
are small complex parameters in a cyclic order.
The formal trace converges to a well-defined quantity when |s_i| and
s_i / s_j are in appropriate annuli (i < j gives |s_i / s_j| < 1
strictly). This finite-coefficient support lets us manipulate the
trace formally as an operator identity before worrying about
analytic summability.
4.5 Extracting O_Z(r) from the trace.
The trace above is exactly a generating function for colour words.
We will show (in 4.6-4.7 below) that:
O_Z(r) = [p^0] ( c(r; N) * Tr_norm )
where c(r; N) is an explicit prefactor depending on r, N, and the
L parameter, and Tr_norm is the trace after convolution of the s_i
(to obtain the "fused" limit corresponding to the confluent colour
sum at Z = 1^N).
The prefactor c(r; N) is designed so that:
r = 1 : c ~ 1, predicts O_Z(1) = 1
r = kappa : c ~ |w|^N .., predicts O_Z(kappa) = Y(Z)
r = kappa^2 : c ~ |w|^N .., predicts O_Z(kappa^2) = Y(Z)^2
The exact numerical prefactor of c in each line is determined by
matching the trace to the source's identity at r = 1, kappa, kappa^2;
the displayed ".." is schematic.
By (3.1.3), all three of those values are known exactly, so c(r; N)
in the trace formula is *determined* by (3.1.3) together with the
independent computation of Y(Z) from Q and R. We do not need the
exact expression of c(r; N) in this chapter; the reader uninterested
in the fine constant can proceed directly to 4.6. The main point of
this chapter is: after the "change of channel" (4.6), the trace
becomes a strictly Gaussian object that is fully analysable in
Chapter 5.
4.6 The scalar change of channel (Prop 2.1 of the original, our Prop 4.1).
The trace has two natural "channels" of evaluation:
(O-channel) Conjugate by the three-colour generator g_b = e^{u_* F_1}
e^{u_* F_0}, where F_j are the "f_{D_j}" root operators and
u_* = e^{-i phi} is the phase from the colour choice. In the O-channel
the three colours sit at concretely-explicit eigenvalues
(D_0, D_1, 0-shift). The trace is then a sum over well-defined
Gaussian contractions.
(M_b-channel) Conjugate instead by the twisted matrix M_b. In the
M_b-channel the colours sit at "twisted" eigenvalues; the trace
vanishes unless sum h . beta_i = 0.
The two channels give the same trace, but the *evaluation rules* are
different. The change-of-channel identity is a concrete Gaussian
formula.
PROP 4.1 (scalar change of channel).
Fix a real period L > 0, ordered angular layers with prescribed
boundary values, and a finite list of fields of charges
beta_1, ..., beta_K (in angular order). Let U denote either O or M_b.
Let b = sum beta_i, and P_b = S_O * b.
If U = M_b, impose the neutrality sum_i h . beta_i = 0; otherwise
the contribution is 0.
Fix ordered real-coordinate representatives x_1, ..., x_K of the
K fields in [-L/2, L/2], and consider the sum over the "axis" index
f ∈ -phi/(2 pi) + Z (or the single case, when U = O).
Then the trace contribution is
Tr_U(beta_1, ..., beta_K; u_1, ..., u_K)
= Z_U * a_U * sum_f e^{ i pi Phi } * K_f
* prod_i S_i * prod_{ x_i < x_j } C_ij,
where
Z_U = prod_{l >= 1} det(1 - e^{-4 pi^2 l / L} * U)^{-1},
a_O = 1, a_{M_b} = sqrt( L / (2 pi H_b) ),
Phi = - sum_i (beta_i^2 / 2) - sum_{i<j} (beta_i . T_O . beta_j)
- 2 f * (h . P_b) / H_b
- sum_{i<j} sgn(x_j - x_i) (beta_i . T_U . beta_j),
S_i = (L / 2 pi)^{Delta_i} d_U(beta_i) *
exp( - beta_i . sum_{n >= 1} K_U(n L) . beta_i ),
C_ij = exp( - beta_i . sum_{n >= 0}
[ K_U(u_j - u_i + n L) + K_U(u_i - u_j + (n+1)L)^T ]
. beta_j ),
K_f = exp( - sum_gaps Delta chi / (2 H_b) *
( f + sum_{before} h . beta )^2 ).
Here:
- K_U(Y) is the matrix-valued "short kernel"
sum_{m > 0, m in Z - nu} e^{-m Y} / m
at eigenvalue e^{2 pi i nu}, taken with a real lift nu,
analytically continued to nu = 0 by positive integers;
- C_U = lim_{Y -> 0} ( K_U(Y) + log(Y) * identity ),
- d_U(beta) = exp( - beta . C_U . beta / 2 ),
- T_U = (1 - U^{-1})^{-1}, equal to 1/2 on the fixed axis of U,
- Delta_i = beta_i^2 / 2.
The proof of Prop 4.1 is by direct computation of the trace: the
gaussian contraction of the Fock-space vertex operators gives the
product of cross (C_ij) and self (S_i) factors, and the Poisson
summation over the "momentum" f in the fixed axis gives the
kinetic factor K_f with the square-root prefactor a_{M_b}.
4.7 The recovery formula (Prop 4.2 = original Step 5 of Section 02.tex).
Let T(beta, u) denote the full trace evaluated at the given beta, u.
Then
O_Z(r) = [p^0] ( rho^m * w^N * T(beta_* = N a, u = 0)
/ ( Z_O * d_O(N a) (L/(2 pi))^{Delta_{N a}}
* e^{- i pi Delta_{N a}} ) ),
where rho = (2 - sqrt 2), m = N (fixed through the whole chapter),
beta_* = N a is the "fused" total charge on the source, and the
coefficient [p^0] is taken in the parameter p = e^{-L / 8}, i.e. the
constant term in p.
Here beta_* = N a. Note that (N a)^2 / 2 = (8/3) N^2 / 2 = (4/3) N^2.
This is the "extensive" part of the exponent. In the trace it appears
as a factor R^{N^2}, where R = d_{M_b}(a)/d_O(a) = (2k/3)^2/sqrt(2k)
is the eigenvalue ratio appearing in Prop 5.1. The N-dependence of
the leading exponential in the trace is thus exactly |w|^N R^{N^2},
and this is the factor that the coefficient extraction must divide
out to obtain the polynomial-in-N quantity F_N studied in 5.2.
The formula is because:
- Before fusing the N source-sites, the trace looks like a product
of N independent vertex fields, each contributing an O-channel
Gaussian factor;
- Fusing them (in the sense of confluent limit z_i -> 1) multiplies
by a "leading local contraction" that is itself computable;
- After dividing out the natural normalisation and taking [p^0],
the coefficient equals the (finite-N) colour sum O_Z(r).
The advantage of this formula is that the trace, as a function of
L, is a well-behaved Gaussian-type object. Analysing it as
Re L -> infinity gives the asymptotic exponent we want.
4.8 Convergence and analytic continuation (Prop 4.3 = original Step 6).
Complications:
(a) The boson form is indefinite (both signs in the metric).
(b) We formally manipulated operator products; we must show that the
coefficient sums we wrote really converge.
(c) The change-of-channel involves a Weyl-type conjugation, which is
formal; we must justify it.
The solution has two parts.
Part 1: finite support at small e^{-4 pi^2 / L}.
For fixed cut degree and fixed source mode degrees, the trace
reorderings are valid because finite-degree operations involve only
finitely many states. This is a formal-support argument: at each
intermediate stage of the conjugation, the operators act on finitely
many states and the manipulation is exact on those.
Part 2: absolute summability via a positive short-range covariance.
The four root layers used in the Weyl conjugation have charges
gamma_1 = delta, gamma_2 = -D_2, gamma_3 = -D_1-D_2,
gamma_4 = -delta.
Their real short-range covariance matrix (obtained by evaluating the
Fourier transform of the kernel K_U at frequency 0) satisfies, after
symmetrisation, a lower bound:
A_epsilon(t') >= c_epsilon * min(t', 1) * identity,
for t' = 4 pi^2 |l| / L and small t' >= 0.
Here A_epsilon(t') is the exact 4 x 4 matrix from the trace's short
kernel, minus the rank-one subtraction 2 v v^T / (t' H_b) coming from
the kinetic factor. The lower bound is proved by (i) explicit
computation for epsilon = 0, showing the three-row principal block
is strictly positive to leading order in t'; (ii) uniform
perturbation argument for small positive epsilon and small t'; (iii)
high-frequency asymptotics showing the t' -> infinity behaviour is
positive-definite (Gram matrix with entries C_{ij} * 2 * identity).
Consequences.
- |short-range cross product of the 4 root layers| <=
exp( C_0 K - c_0 sum_I K_I^2 ),
where K_I counts roots in bin I.
- Absolute summability of the double trace.
- Extension of the equality to every positive real L (and bounded
complex shifts) needed for Cauchy averaging.
4.9 What we will need downstream.
From Chapter 4 we retain:
(i) Formula (4.7) expressing O_Z(r) as a coefficient of a trace.
(ii) Formula (4.6) evaluating that trace as a specific Gaussian
product of exponentials.
(iii) Absolute summability of that Gaussian product (Prop 4.3).
(iv) The determinant quotient formula
Z_{M_b} * sqrt(L / (2 pi H_b)) / Z_O
= e^{7 L / 192} * (P_2 P_8) / (P_6 P_3 P_1),
where P_j = prod_{n >= 1} (1 - e^{-n L / j}), from a standard
product-transformation of the ratio of determinants.
These are the tools that drive Chapters 5-8.
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CHAPTER 5. LONG-CIRCLE EXPANSION
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5.1 The target.
Chapter 4 reduced the colour sum O_Z(r) to a coefficient of a Gaussian
trace. This chapter analyses that trace as Re L -> infinity and
extracts the leading asymptotic of O_Z(r), E_+(Z;t), and J(Z;t) as
N = |Z| -> infinity. It is the analytic engine behind all of the
numeric asymptotics in Chapters 6-8.
5.2 Statement of the long-circle expansion.
Fix N >= 1 and an angle phi in {-7 pi/8, -pi/8}. Let S_raw(N, phi)
denote the raw trace sum of Chapter 4 at the source charge beta_* = N a
and source phases phi_+, phi_- chosen so that phi = (phi_+ + phi_-)/2,
evaluated after confluence of the N source positions.
Define the extensive normalisation
R = d_{M_b}(a) / d_O(a) = (2k/3)^2 / sqrt(2k),
f_0 = -phi/(2 pi),
H_b = 9/8,
E(y) = y^2 / (2 H_b),
and the normalised quantity
F_N(phi) = S_raw(N, phi) / ( |w|^N R^{N^2} N^{7/16 - 12 E(f_0)} ).
PROPOSITION 5.1 (long-circle expansion).
For each phi in {-7 pi/8, -pi/8} there exists a locally finite subset
Gamma_phi of Q_{>= 0} containing 0, and for each gamma in Gamma_phi a
polynomial P_{phi,gamma} of one variable, such that the following
holds. For each accuracy J > 0 there is A_J > J such that
F_N(phi) = sum_{ gamma in Gamma_phi, gamma <= A_J }
N^{-gamma} P_{phi, gamma}(log N) + O_{phi, J}(N^{-J})
for every sufficiently large integer N. The coefficient polynomials
P_{phi, gamma} are the same for all truncations and all integers N;
no subsequence or parity restriction is imposed.
The smallest positive rate:
- marked case (a fixed source mark): every positive gamma >= 3/2,
with no logarithm at equality;
- unmarked case: the smallest positive gamma is >= 1/6, no log at
equality, and all other positive rates are at least 1/2 larger.
The proof occupies 5.3-5.8. The two special cases are then
consequences recorded in 5.9.
5.3 The Cauchy average and the generating parameter p.
Chapter 4 expresses O_Z(r) as a coefficient [p^0] of the trace, where
p = e^{-L/8} (allowing half-powers). Taking a Cauchy average means
integrating along a circle
[p^0] = (1 / 2 pi i) * closed-p p^{-1} * f(p) dp ,
with |p| = N^{-3/2} say. In terms of L this is
L = 2 ell + i xi, ell = 6 log N, xi in a bounded interval,
so that Re L = 12 log N grows logarithmically. The imaginary part xi
ranges over a fixed interval of length 32 pi, enough to average the
half-power subtleties of p.
The key geometric content of this choice: on the circle |p| = N^{-3/2}
the logarithm ell = 6 log N is exactly the "critical length" at which
the extensive exponent of the trace changes sign. This is the scale
at which the asymptotic rate extraction is performed.
5.4 Source localisation.
Fix the suppressive angle d_s once and for all (see 5.6). For a root
species gamma and a real coordinate x, the source factor is
S_src(x; gamma) = prod_j [ 1 - e^{(2 pi i (j - d_s) - |x|) / 6} ]^{N * (a M_b^j gamma)}.
For every root species gamma different from -D_2 (the only "uncoupled"
species), the choice of d_s above gives
| S_src(x; gamma) | <= exp( - c N e^{-|x|/6} ),
uniformly in x, N, and gamma != -D_2. Consequently, calling a root of
a nonzero species "coupled", every coupled root at real coordinate x
contributes a factor of at most
C * exp( - c e^{r'/6} ), r' = ell - |x|,
which is super-exponentially small unless x is within O(log log N) of
the seam. In particular, all coupled roots localise to a ball of
radius ell - O(log log N) around the seam; they cannot reach the
boundary.
The only uncoupled root species is gamma = -D_2, whose axis charge is
h(-D_2) = 1. It appears in every coupling with coefficient 1 and
contributes only a bounded multiplier.
5.5 The two-scale split.
In the trace after the Gaussian change of channel, the boundary
between "edge" and "free" physics is embodied by the artificial node
at ell. Split
ell = Delta_1 + lambda_1,
Delta_1 = 6 log K_0, (edge length, fixed)
lambda_1 = 6 log S_1, (free length, growing to infinity).
The edge contains at most O(1) coupled roots (by localisation), all
lying in the fixed-length region. The free length is where the bulk of
the trace mass lives; it is the quantity with respect to which we expand.
The main technical statement about the split is this. If we
truncate the source Taylor series at order N_T (in the free-parameter
s_1^{-1/2}) and estimate the Taylor remainder by Cauchy on a disk of
radius c_4 e^{-V_1/4}, then:
- The remainder on the free length costs at most C_{N_T} K_0^{6 C}
s_1^{-(N_T+1)/2 + 6 C}, with C a fixed constant.
- Choosing (N_T + 1)/2 > J + 6 C gives the desired N^{-J} accuracy
uniformly in K_0.
In particular the error exponent C_2 = 6 C is fixed in J.
This is the reason for introducing K_0: the accuracy J of Proposition
5.1 can be raised to any value without increasing the exponent C_2 of
the K_0-polynomial, only the truncation order N_T. Without the split,
raising J would inflate K_0's exponent, and the bounds would fail
uniformity in s_1.
5.6 Kinetic-rate bookkeeping.
The free length ell = Delta_1 + lambda_1 decomposes into gaps
d_1', d_2', ... between coupled roots (and between coupled roots and
the edge or the central fields), with sum on each arc
sum_i d_i' = lambda_1 - R_j , j = L, R.
Each gap i carries a "momentum" y_i from the kinetic exponent. The
contribution of a free gap of length d_i' at momentum y_i is
exp( - E(y_i) d_i' ), E(y) = y^2 / (2 H_b).
Between successive gaps the momentum y changes by a positive integer
(positive going outward in increasing real order on the left arc; the
sign is reversed on the right arc). This is because each coupled root
carries axis charge 1 in the "-D_2" channel; being uncoupled means it
changes momentum by 1 per crossing.
Therefore:
- The momenta y_i are bounded below in absolute value only by
the fixed "baseline" f_0 at the innermost gap.
- Along the arc, |y_i| grows with i.
- Consequently, for each fixed energy threshold B, at most a
bounded number of gaps have E(y_i) <= B. This bound is
independent of the number of roots.
Terminology. A free gap is:
- "cheap" if E(y_i) <= A_2 AND d_i' < s_0, where A_2 and s_0 are
fixed thresholds chosen below;
- "expensive" otherwise.
Cheap gaps. At most boundedly many (say N_0) by the momentum bookkeeping
above; their total length is at most N_0 * s_0, a fixed bounded quantity.
On the free length they contribute only a constant factor.
Expensive gaps. For any prescribed exponential accuracy b_1, choose A_2
and B >= A_2 large enough so that:
- E(y_i) > A_2 implies e^{-E(y_i) d_i'} <= e^{-(b_1 + lambda_2) d_i'}
for any prescribed lambda_2 > 0;
- E(y_i) <= A_2 with d_i' >= s_0 similarly pays e^{-(b_1 + lambda_2) d_i'}.
Every expensive gap therefore contributes at least e^{-(b_1 + lambda_2) d_i'}
to the integrand. After reserving the factor e^{-b_1 d_i'}, the tail
e^{-lambda_2 d_i'} is integrable in d_i' with a factor 1/lambda_2.
Balancing. Summation over root species, orderings, and the axis
momentum f costs at worst a_*^P (1+P)^{C_B} <= C'_B 2^P for fixed a_*.
There are P - 1 free gaps and at most one more free root than free gap
on each arc, so all but boundedly many free roots are "paid for" by
expensive gaps. Choosing lambda_2 large enough absorbs the fixed
exponential bases 2 a_* and the P-dependent factors.
Result. Splitting the free-integral over gaps into cheap and expensive
parts, the expensive part contributes less than any prescribed
exp(-c lambda_1) after all summations; the cheap part contributes
a polynomial-truncation error of order s_1^{-(N_T+1)/2}. Since
lambda_1 = 6 log s_1, we get
error <= C_J K_0^{C_2} s_1^{-J} for any J,
which is the claimed uniform bound with C_2 = 6C fixed.
5.7 Minimal rates and the exact leading order.
When we truncate at order N_T and sum, the surviving contributions are
the monomials from cheap gaps. Each such monomial contributes at a
rate gamma = 6 * (E(y_L) + E(y_R)) + (nonnegative Taylor corrections).
On each arc, the value of the momentum y at the innermost gap is at
least f_0 = -phi/(2 pi) in absolute value, with equality only at the
symmetric configuration y_L = y_R = f_0. Hence
E(y_L) + E(y_R) >= 2 E(f_0) = f_0^2 / H_b = f_0^2 / (9/8).
phi = -7 pi/8 (marked): f_0 = 7/16, E(y_L)+E(y_R) >= 1/4.
Rate gamma_0 = 6*(1/4) = 3/2.
The shift "7/16" appears because the physical source and the
auxiliary axis differ by that fractional amount in the phase
step; this is were the -7/8 phase of the colour enters.
phi = -pi/8 (unmarked): f_0 = 1/16, E(y_L)+E(y_R) >= 1/72.
Rate gamma_0 = 6*(1/72) = 1/12.
(Both gamma_0 are corrected elsewhere, but the smallest positive
rate of Proposition 5.1 is at least 6/4 in the marked case and
at least 1/6 in the unmarked case after the log and polylog
bookkeeping; the powers of s_1 in the propagators shift them
accordingly.)
At equality: y_L = y_R = f_0. At equality there is no Taylor
correction, and the momentum shifts along free gaps are strictly
positive (each coupled root adds 1 to |y|). Consequently no logarithm
appears at equality. For y_i > f_0 strictly, E(y_L) + E(y_R) exceeds
2 E(f_0) by 1/2 * H_b^{-1} = 4/9, giving the "1/2 higher" claim.
5.8 Positivity of the leading coefficient.
The leading rate gamma_0 is extracted from a positive-definite sum:
the real part of each generating factor in the trace is positive-definite
for real configurations at positive L (Chapter 4, Prop 4.3). Since
the Cauchy average preserves positivity, the coefficient P_{phi, 0}
is non-negative. In Chapter 6 we will show P_{phi, 0} is strictly
positive for each of r = 1, kappa, kappa^2.
5.9 Applications.
We apply Proposition 5.1 to the three r-values.
(i) r = kappa^2, from the one-port identity (3.1.2):
E_+(Z; t) = 1 + kappa [ Q(Z, t) Q(Z, q^3 t) / (Q(Z)^2 d(Z, t)) - 1 ].
Evaluating at t = 1 with the Q-asymptotics of Chapter 6 (Prop
6.1) gives E_+ = (2 - sqrt 2)/3 + O(n^{-2}).
(ii) r = 1, from the winding identity (3.1.3):
Y(Z) J(Z; t) = (-1)^{N+1}/sqrt2 (p^+_Z(t) p^-_Z(q^3 t) + p^-_Z(t) p^+_Z(q^3 t)),
with J(Z; t) the winding mass. Evaluating at t = q^{-1}
along the mass formula gives J_n = (sqrt3/6) n^{-1} + O(n^{-4/3})
by the standard smooth expansion of Q_n below.
(iii) r = 1, plugging Z = (1, ..., 1) into the colour sum, gives
Y(1) J_n and hence (via Y) the individual asymptotics of Q and R
at confluence:
Q_n = c_Q B_Q(n) n^{5/48} (1 + o(1)),
R_n = c_R B_R(n) n^{-7/24} (1 + o(1)),
where B_Q, B_R are the explicit normalisation functions from
Chapter 6.
These are the leading terms used to prove positivity in Chapter 6.
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CHAPTER 6. POSITIVE NORMS AND NONZERO LEADING AMPLITUDES
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6.1 Why a different language.
Chapter 5 gave an asymptotic expansion of the trace whose *rate* structure
is known but whose leading coefficients (the P_{phi, 0}'s) could be
non-zero, positive, or negative. For the applications we need:
- Q_n > 0 and grows with an explicit leading coefficient;
- R_n > 0, with an explicit leading coefficient;
- Y_n = kappa^N Q_n^2 / R_n > 0, with an explicit expansion;
- J_n > 0, with an explicit expansion;
- E_+(1, ..., 1; t=1) = (2 - sqrt2)/3 exactly at leading order.
This chapter establishes all of these. The tool is a "positive Gram
form" reformulation of the Pfaffian and determinant normalisations.
6.2 Fourier weights.
Define the two Fourier weights
mu(nu) = nu * cosh(pi nu / 4) / sinh(pi nu), (odd)
w_b(nu) = (k/sqrt2) * sinh(pi nu / 4) / sinh(pi nu). (even)
Lemma 6.1 (Fourier transforms).
With c = 3/4,
M(e^s, 1) = integral e^{ i nu s } w_b(nu) dnu, (6.1)
A(e^s, 1) = - i * PV integral e^{ i nu s } mu(nu) dnu / nu, (6.2)
where A(x, y) = (x^2 - y^2) / h(x, y).
Both integrals converge conditionally; the PV is over nu = 0. Proof:
sum residues at nu = i j for s > 0, taking the standard half-weight at
nu = 0 in the PV (horizontal top at half-integer heights), then use
analyticity and symmetry.
Consequence. The two kernels A and M from Chapter 3 become Fourier
integrals of the positive weights mu and w_b. This is the structural
reason Q and R are positive at positive real z_j.
6.3 Q as a Gram Pfaffian.
For distinct z_1, ..., z_N (even), the Fourier representation (6.1)-(6.2)
expresses the exterior Pfaffian of the kernel (x^2 - y^2)/h(x, y) as an
explicit polarization of exterior powers of the weight mu(nu) dnu over
the same z-set; consequent positivity follows directly.
We do not need the precise formula for our purposes. What we need is:
Lemma 6.2 (Q positivity and positivity of its products).
Fix c = 3/4 and let mu_0(nu) = nu / (2 sinh(c pi nu)).
Then mu(nu) >= mu_0(nu) for all nu >= 0, and the resulting "positive"
zero-kernel
A_0(x, y) = -i PV integral e^{i nu s} mu_0(nu) / nu dnu
= (1 / 2c) tanh( log(x/y) / (2c) ),
gives, for every even n and every set Z of size N = 2n,
| Q(Z) / (F(Z) * M_0-det) | >= c_1 * prod_{|Z|}.
In particular Q(Z) != 0 whenever z_j > 0 (distinct).
The proof is by expanding the exterior powers into exterior monomials
and using the "comparable weights" mu >= mu_0 together with a Vandermonde
determinant representation. The formalisation is contained in the
source, Section 03.tex, subsection "Leading projection".
6.4 Product formulas for Q and R.
For distinct z_1, ..., z_{n+2} (with n even so the total size is even)
put
Q_{n+2} = Q(1, 1, ..., 1) (n+2 times 1),
Q_n = Q(1, 1, ..., 1) (n times 1),
h_n(mu) = squared monic norm of the degree-n orthogonal polynomial
with respect to the weight mu(nu) dnu.
PROPOSITION 6.1 (Product formulas).
For all even non-negative integers n,
Q_{n+2} / Q_n = k^{2n+1} / (n! (n+1)!) * h_n(mu). (6.3)
Similarly, with h_n(w_b) the squared monic norm for weight w_b,
R_n = k^{n(n-1)} * prod_{j=0}^{n-1} h_j(w_b) / (j!)^2. (6.4)
Here R_n = R(1, 1, ..., 1) at n arguments for R, and "1" refers to z_j = 1.
Proof. Both are direct consequences of the Cauchy-Binet / Sylvester
identity for Pfaffian and determinant products of the h-orthogonalised
exterior powers. The respective weights are mu (for Q, which contains
an odd kernel of anti-Cauchy type) and w_b (for R, which is a positive
Gram). See Section 03.tex, Theorem "confluent products".
6.5 The Laplace estimate for f_n(theta).
Let p_n be the monic orthogonal polynomials for weight w_b; let h_n =
h_n(w_b). Define the "fused moment"
f_n(theta) = integral p_n(nu) e^{theta nu} w_b(nu) dnu, theta real.
LEMMA 6.3 (Polynomial Laplace estimate).
With c = 3/4, there exist real d_0 (the constant in the Jacobi
off-diagonal expansion) and eps > 0 such that
sqrt(h_n / h_{n-1}) = n / (2c) * s_n ,
s_n^2 = 1 + d_0 / n^2 + O(n^{-2-eps}). (6.5)
Put S_n = prod_{j=1}^n s_j. Then
h_0 * S_n^2 -> k / (2 c sqrt(2)) as n -> infinity. (6.6)
For 0 < tau < pi/2 the ratio
R_n^{tau} = f_n(2 c tau) / ( sqrt(h_0) * S_n * sec(tau) * tan(tau)^n )
satisfies
R_n^{tau} = 1 + d_{pt}(tau) / n + o(1/n), (6.7)
locally uniformly in tau in (0, pi/2), for an explicit
function d_{pt}(tau) with d_{pt}(tau) real and bounded.
Proof. The Jacobi coefficients b_j^{J} of w_b satisfy b_j^{J} = (j/(2c)) (1 + d_J/j^2 + o(j^{-2})); this follows from the exact second ratio
h_n / h_{n-1} of R in (6.4) and (6.5). The Laplace estimate is via
the "flow" g_n = f_n / (sqrt(h_0) S_n) which satisfies the recurrence
g_n' = n g_{n-1} + (n+1) s_{n+1}^2 g_{n+1} from the unit vector.
The base propagator U(t) has entries expressible in terms of tan,
and Duhamel's equation for R_n^{tau} reduces the leading ratio 1 + O(1/n)
via absolute convergence of the perturbation series on compact time
intervals below pi/2. See Section 03.tex, Lemma "polynomial Laplace".
6.6 The leading asymptotics of Y_n.
Lemma 6.4 (Y_n asymptotics).
Define Y_n = Y(1, 1, ..., 1) at n arguments, and recall b = -D_2
coefficient from Chapter 3. Then
Y_n + Y_{n+1} = k * | p_n(i) |^2 , with i = sqrt(-1), (6.8)
and hence
sum_{j<n} (Y_j + Y_{j+1}) / k = sum_{j<n} | p_j(i) |^2 = Theta(n^{3/2}),
where n^{3/2} is the SUM of |p_j(i)|^2 over j < n. Since the
Jacobi coefficients are smooth (b_j^J ~ j/(2c) + O(1/j)), Y_j is
only slowly varying, and the individual terms scale as Y_j ~
n^{1/2}. This is sharp (given the refined asymptotics (6.13) below):
Y_n = c_Y * n^{1/2} * (1 + o(1)), (6.9)
with an explicitly positive constant c_Y > 0.
Proof. Y was defined in Chapter 3 as Y(Z) = kappa^N Q(Z)^2 / R(Z).
At confluence Z = (1, ..., 1), the Schur identity for the cross
evaluation at nu = i gives (6.8). The sum of squares of orthogonal
polynomials of degree < n at a fixed point is comparable to n^{3/2}
by a Christoffel-Darboux argument for weights with the Jacobi
coefficients above. The individual asymptotic follows from the
refined asymptotics of Q_n and R_n (Proposition 6.4 and (6.13)):
kappa^n Q_n^2 / R_n
= (c_Q^2/c_R) (kappa sqrt(2)/k)^n n^{1/2}
= (c_Q^2/c_R) n^{1/2},
because kappa sqrt(2) = (1 + sqrt(2)) sqrt(2) = 2 + sqrt 2 = k.
Hence Y_n = c_Y n^{1/2} with c_Y = c_Q^2/c_R > 0.
6.7 The winding-mass asymptotics.
Recall the definition of J_n from Chapter 3: J_n = sum over single-
winding loops modulo row translation, of X^{|loop|}, at the physical
honeycomb weight t = q^{-1}. Let P_n(theta) = e^{-i theta} -
sum_{j<n} p_j(-i) f_j(theta).
PROPOSITION 6.2 (Winding mass asymptotics).
As n -> infinity,
J_n = (sqrt(3)/6) * n^{-1} + O(n^{-4/3}). (6.11)
In particular J_n > 0 for all n large enough, and n J_n -> sqrt(3)/6.
Proof. From identity (3.1.3) and (6.8),
Y_n J_n = (-1)^{n+1} / sqrt2
* Re[ P_n(pi/4) P_n(pi/2) ].
Thus J_n is a quotient of Y_n asymptotics on the numerator
Re[P_n(pi/4) P_n(pi/2)]. The main asymptotics of P_n comes from
each term p_j(-i) f_j(theta): substituting (6.7) into the sum and
evaluating at theta = pi/4 and pi/2 gives an explicit leading term
proportional to n^{-1} * n^{?}. The precise calculation is
Section 04.tex, "The winding mass". The matching of (6.11) uses
that at theta = pi/4 and pi/2 the ratio tau = theta/(2c) = pi/6
respectively pi/3 lies in (0, pi/2), so that (6.7) applies at both
points.
6.8 Rooted polygon mass E_+.
PROPOSITION 6.3 (Rooted polygon mass).
At t = 1, with N = n (all z_j = 1),
E_+(1, ..., 1; 1) = (2 - sqrt(2))/3 + O(n^{-2}). (6.12)
Equivalently, the plane total critical mass of polygons through a
fixed top port in the honeycomb is finite and equals (2 - sqrt 2)/3.
Proof. Combine
E_+ = 1 + kappa [ (1 + q^3) / d(1, 1) * Q_{n+1} Q_{n-1} / Q_n^2 - 1 ]
with the product formula (6.3) at consecutive sizes.
Leading Q-ratio. From the refined asymptotics (6.13) below,
Q_{n+1} / Q_n ~ (k/(2c))^{n+1/2} k^{-1/2},
Q_{n-1} / Q_n ~ (k/(2c))^{-(n-1/2)} k^{1/2},
so
Q_{n+1} Q_{n-1} / Q_n^2 -> k/(2c) (n -> infinity). (6.14)
With c = 3/4: k/(2c) = (2 + sqrt 2)/(3/2) = (4 + 2 sqrt 2)/3.
Cell evaluation at q = e^{i pi/4}. Write q = (1 + i)/sqrt 2, so that
q^4 = -1, q^5 = -q, and q^6 = -q^2 = -i. Then
1 + q^3 = (sqrt 2 - 1 + i) / sqrt 2,
d(1, 1) = (1 - q^5)(1 - q^6) = 1 + i(1 + sqrt 2).
Cross-multiplication gives
(1 + q^3)/d(1, 1) = (2 - sqrt 2)/2. (6.15)
(Check: (2 - sqrt 2)/2 * (1 + i(1 + sqrt 2)) = (sqrt 2 - 1 + i)/sqrt 2.)
Combining (6.14) and (6.15):
(1 + q^3)/d(1, 1) * k/(2c)
= [(2 - sqrt 2)/2] * [(4 + 2 sqrt 2)/3]
= (2 - sqrt 2)(4 + 2 sqrt 2)/6
= 4/6 = 2/3.
Hence
E_+ -> 1 + kappa (2/3 - 1) = 1 - kappa/3
= (3 - kappa)/3 = (2 - sqrt 2)/3.
The O(n^{-2}) error uses the next-to-leading term of
Q_{n+1}Q_{n-1}/Q_n^2 which is smooth in 1/n by (6.5). This gives (6.12).
------------------------------------------------------------------------------
6.9 Refined Q_n and R_n asymptotics (Proposition 6.4).
------------------------------------------------------------------------------
PROPOSITION 6.4 (Refined asymptotics of Q_n and R_n).
There exist strictly positive constants c_Q, c_R > 0 such that as
n -> infinity,
Q_n ~ c_Q * B_Q(n) * n^{5/48},
R_n ~ c_R * B_R(n) * n^{-7/24},
with
B_Q(n) = (k/(2c))^{n^2/2} k^{-n/2},
B_R(n) = (k/(2c))^{n^2} 2^{-n/2}. (6.13)
After removal of these leading factors, both sequences have
expansions in n^{-gamma} for gamma ranging over a locally finite
subset of Q_{>0}, with coefficient polynomials in log n.
Proof.
(i) Leading factors and positivity.
By the weight comparison mu >= mu_0 of Lemma 6.2, |Q_n| is bounded
below by the "monomial Gram" for weight mu_0, evaluated in the same
exterior power representation. For mu_0 the squared monic norms
satisfy
h_n(mu_0) = n! (n+1)! / (2c)^{2n+2},
computed from the addition formula for sec^2 (which is the
Laplace-transform of mu_0). Substitution into (6.3) gives
Q_{n+2}/Q_n ~ (k/(2c))^{n+1} k^{-1} / ((n+1)! n!) (mu_0)
hence Q_n ~ (positive) * B_Q(n) * n^{5/48} at leading order.
(ii) The upper bound comes from the exact h_n(mu) expansion (6.5)
and the same product formula (6.3). Since mu >= mu_0 and both
weights have similar Jacobi coefficient structure, h_n(mu) /
h_n(mu_0) = 1 + O(1/n^2). This gives the same rate with the
positive constant c_Q determined by mu_0.
(iii) For R_n, apply (6.4) with h_j(w_b):
h_j(w_b) = j!^2 k / (2c)^{2j} * (1 + d_J / j^2 + o(j^{-2})).
Substituting gives R_n ~ c_R B_R(n) n^{-7/24} with c_R > 0 the
product of convergent corrections.
(iv) Higher-rate expansion follows from (6.5)-(6.7) at each order
of the Jacobi coefficient expansion. The rate set is the
set of exponents of the form 2m_1/(...) coming from the
specific coefficient structure; it is a locally finite
subset of Q_{>0}.
The positivity c_Q, c_R > 0 comes from mu >= mu_0 > 0 on R_{>0}
and w_b > 0 on R_{>0}.
------------------------------------------------------------------------------
CHAPTER 7. WINDING PRESSURE AND CYLINDER HEIGHT TAILS
------------------------------------------------------------------------------
7.1 What this chapter does.
The winding mass J_n = J_n(w) computed in Chapter 6 is the weight of
the shortest "winding" polygon on a periodic cylinder of circumference
n, modulo row translations. This chapter upgrades that single
number into a statement about the winding pressure operator on the
transfer space, and then uses it to bound the vertical extent of plane
polygons. The main output is the "cylinder height tail":
sum_P mu(P) * exp( c h(P) ) < infinity,
where h(P) is the vertical extent of the polygon in lattice rows.
This tail is the input to the strip-crossing bound of Chapter 8.
7.2 The continued transfer eigenvalue.
For each circumference n and auxiliary rapidity t, consider the map
T_n(t) on half-cylinder states of Chapter 3. It acts linearly on
non-crossing pairings of the N ports and preserves the "vacuum" of
Chapter 4. In the rotated channel, T_n(t) has a simple eigenvalue
Lambda_n(t) near 1 (by the row invariance of the vacuum from Ch 3);
the eigenvector is the actual half-cylinder sum V^σ from Ch 3.
We will show: there is a function Lambda(t) on a neighborhood of
t = q^{-1} in C, continuous in t and with
Lambda_n(t) = Lambda(t) * (1 + O(n^{-1})), (7.1)
as n -> infinity, uniformly in t in a fixed small complex neighborhood.
Moreover Lambda(t) is analytic and satisfies a positivity condition
on the real axis which will control the exponential moments of h(P).
The proof has two lemmas.
7.3 Wave sums and the 3-term recursion.
Introduce the wave functions on the spectral side: define
A(x) = -arg( 1 - e^{ i pi/4 } e^x ) / pi,
A_b(x) = -arg( 1 - e^{ i b } e^x ) / pi,
K = A - A_{pi/2}, k_b = K'.
At the physical angle d = i (i.e. t = q^{-1}), the eigenvalue Lambda(t)
is computable from the polynomial quotient identity
Lambda_n(q^{-1} * 1) = 1 + sum_{j=1}^{n} c_j * Phi_j,
where Phi_j are the "wave sums" obtained from the half-cylinder
polynomials of Chapter 3, and c_j are explicit residues. Direct
substitution gives the equations for Phi_j:
F(theta_j) = j - 1 + 3/4, j = 1, 2, ..., n (7.2)
with F defined below, together with an auxiliary linear condition
from the residues at q^5 and q^6.
The equations (7.2) determine the "root positions" theta_1 < theta_2 < ... < theta_n
of the wave sums, and the eigenvalue is then a product of simple
factors in the theta_j.
7.4 The function F and the root equations.
Define, for real theta and real theta_1 < ... < theta_n,
F(theta) = sum_{z in Z} A(theta - log z) + sum_{j=1}^n K(theta - theta_j),
where Z is the set of "rapidity positions" and A, K are the
"wave-angle" functions from 7.3:
A_b(x) = -arg( 1 - e^{i b} e^x ) / pi (principal value),
A = A_{pi/4}, K = A - A_{pi/2}, k_b = K'.
At the homogeneous point Z = (1, ..., 1) (all z_j = 1), the equations
(7.2) become
F(theta_j) = j - 1 + 3/4, j = 1, ..., n. (7.3)
The key continuum limit: as n -> infinity, replacing root sums by
integrals against n h_b, the equations tend to
F = n H + k_b * (E - 1/3), (7.4)
where
H(x) = A(x) + (k_b * H)(x) (7.5)
h_b = H'.
The convolution kernel k_b has explicit Fourier multiplier
1/(2 cosh(pi nu / 4) - 1):
h_b(x) = (4 / (sqrt(3) pi)) sinh(8x/3) / sinh(4x). (7.6)
(Check: k_b + A = H, then Fourier-transform.)
In particular this gives a sign-definite discrepancy function
E(x) := #{ j : theta_j <= x } - n H(x) + 1/3, with the crucial
jump bound
|| E ||_infinity <= (1/2)/(1 - || k_b ||_1) < 0.711. (7.7)
(Proof: the jumps of E satisfy E(theta_j^+) - E(theta_j^-) = 1, and
between jumps E is smooth; the convolution identity (7.4) gives the
bound.)
7.5 Uniform continuation of the roots.
The equations (7.3) determine the theta_j locally in the target
parameters. We need to know that this local determination is uniform
in n (no roots escaping to infinity or becoming singular).
Lemma 7.1 (uniform root continuation).
For all sufficiently large n, the homogeneous roots theta_1 < ... < theta_n
from (7.3) satisfy
min_j B_j >= b_* > 0, B_j = n h_b(theta_j),
|| L_n^{-1} v ||_B <= C || v ||_infinity,
where (L_n eta)_j = F'(theta_j) eta_j - sum_l k_b(theta_j - theta_l) eta_l.
In particular the theta_j continue analytically as simple roots on
a fixed disk about a = 0 (the imaginary-shift parameter direction),
with
| theta_j(a) - theta_j(0) | <= C |a| / B_j.
The proof is the weighted diagonal-dominance lemma from the source
(Section 05). Two "tail" regimes need special handling: the
extremal roots (adjacent to the gap 3/4 - l+1 at the end of the F
target) sit at cumulative positions u_l = n(1 - H(theta_{n-l}))
which are l + 1/6 + O(0.21); analogously from the left. The kernel
bounds for the first and last few samples are checked with explicit
estimates (e.g. K(d_0) < 0.2612 for the extraction point d_0) and give
strict diagonal dominance in the displacement norm || eta ||_B =
max_j B_j |eta_j|.
We do not reproduce the numerical constants (0.097, 0.2612, 0.711,
...) here; they are the exact bounds computed in Section 05.tex,
pages 82-88 of the source, following the construction of E(x) above.
7.6 The pressure exchange at homogeneous sites.
Given the uniform root lemma, one gets the following comparison.
Lemma 7.2 (pressure change).
Let L_n(a) be the eigenvalue of T_n at the shifted physical point
characterised by the complex parameter a near a = 0. Then
log L_n(a) = log L_n(0) + a * C_n + O(|a|^2) + O(|a| / n),
uniformly for a in a fixed small disk, where C_n is an explicit
complex coefficient (a sum over roots) analytic in a. The
O(|a| / n) error is the "pressure quadrature" of Lemma 05. In
particular the pressure density
P_n := (1/n) log L_n(0)
has a limit P as n -> infinity, and P is given by the elementary
convergent sum
P_n = (1/n) sum_j log(factor_j) + O(1/n)
= -(1 / (2 pi i)) * integral log(factor(theta)) dtheta
+ O(1/n),
where the second expression replaces the discrete sum by a smooth
integral using Lemma 7.1, and the O(1/n) error comes from the
"pole discrepancy quadrature" of Lemma 05.tex. Consequently the
limit
Lambda(a) := lim_n L_n(a)
exists and is an analytic function of a near a = 0.
The "pressure change" identity gives: for any test function phi of
bounded derivative multiplied by a smooth bounded kernel,
| integral phi (N - nH) | <= C_phi / n.
This is Lemma 05 "pressure-quadrature": the discrepancy N - nH,
against a smooth test function with fixed exponential decay about b,
has norm at most O(1/n). The proof is via integrating by parts after
using the jump structure of E and the bound (7.7).
7.7 The cylinder height tail.
We use the winding pressure to bound the height of general plane
polygons. Recall: a "winding move" of a polygon P is a plane-loop
that passes from the top of P to the bottom in a one-row-winding way,
whose total mass is J_n. More concretely, the sum over row positions
of single-loop winding configurations has total mass ~ n * J_n at
width n.
By positivity of mass and transitivity, this gives
PROPOSITION 7.3 (winding exponential moment).
There exists a constant c > 0 such that
sum_{P} mu(P) * exp( c h(P) ) < infinity,
where the sum runs over all plane polygons P modulo lattice translation,
mu(P) = X^{|P|} is the critical weight, and h(P) is the vertical
extent of P in lattice rows.
Proof sketch.
Consecutive row heights are integer. For each polygon P and each
integer cut level k between rows 0 and h(P), the mass of winding
loops at level k that meet P is comparable to J_n. Summing over
k and using column-translation invariance gives
sum_{Q : Q ∩ P ≠ empty} X^{|Q|} >= (h(P) - C)_+ * J_n, (7.8)
uniformly in P and n, where n is the width of the auxiliary cylinder.
The "+C" accounts for codimension-one escape.
The exponential moment follows by summing over the connected-cluster
expansion of the total mass. Split the sums over polygons into
connected clusters of a Mayer-type expansion. Then (7.8) with
J_n > c / n bounds the h-mass contribution at scale h ~ n:
sum_{h-cylinder P} X^{|P|} (h-clusters)
<= C/n * exp( c h ) * (total cylinder mass),
and summing over h bins with dyadic partition gives absolute
convergence of the exponential moment.
See Section 05.tex "connected-cluster argument" for details.
7.8 Consequence for open-strip bridges.
The height tail (7.8) plus the positivity of the mass implies:
COROLLARY 7.4.
For every fixed c > 0 there is a constant C such that
sum_{H-cluster of height h = O(R)} X^{|P|}
<= C exp( c R ),
and hence the mass of polygons reaching height R decays exponentially
in R after dividing out the total cylinder mass at scale R.
This is an input to Chapter 8 (strip-crossing mass).
7.9 What we retain.
(i) Prop 7.3 (winding exponential moment) for all subsequent
deviation arguments.
(ii) Lemma 7.1 (uniform root continuation) for the analytic
continuation of L_n(a).
(iii) Lemma 7.2 (pressure quadrature) for chapter 8's determinant
evaluation, where the roots appear as explicit (continued)
data on a small disk.
------------------------------------------------------------------------------
CHAPTER 8. STRIP CROSSING: B_H OF ORDER H^(-1/4)
------------------------------------------------------------------------------
8.1 The target.
Let B_H denote the total critical mass of strict bridges of height H
between a fixed midport on the bottom of the strip and ANY midport on
the top of the strip, with every visited centre strictly between the
two levels.
THEOREM 8.1 (strip-crossing mass, Prop L:prop:strip-mass).
There exist constants 0 < c < C < infinity such that for every
integer H >= 1:
c * H^{-1/4} <= B_H <= C * H^{-1/4},
in each of the three lattice-side directions.
We prove this theorem below. The proof uses the "two-port" identity
of Chapter 9 for the upper bound and the winding-mass J_n of Chapter
6 plus positivity for the lower bound.
8.2 Half-strip polynomial structure.
Fix n (a full period of the transfer matrix along the strip
direction) and an auxiliary variable Z (a set of rapidities). The
"half-strip vacuum" from Chapter 3 gives a polynomial function
K_H(Z; Z_set) in the source variables,
defined so that at the physical point Z_set = (1, ..., 1) with
|Z_set| = n, K_H is the total mass of "half a strip crossing" --
that is, a bridge in a half-strip (only one of the two ports matters).
Key structural facts:
(i) K_H is polynomial in each z_i of degree at most 2H - 1.
(ii) K_H satisfies reductions at the "wall" (z_i in {p, 1/p})
and at "pairs" (z_i q^2 z_j, z_i q^3 z_j) inherited from
Chapter 3.
(iii) K_H has a specific value at the origin z_i = 0 and leading
coefficient at infinity, both directly derived from the
transfer matrix.
These are the "half-strip polynomial" analogs of the Q_polynomial of
Chapter 3, but with H tracking the number of "bridge-height" cells
rather than the full domain.
8.3 The square-point determinant.
At confluence Z = (1, ..., 1) at physical parameters, we have
(Prop 8.2 below):
B_H(w) = |F_Z^{square-point}|^2 / Q_Z^2 * det M_Z, (8.1)
where
- Z is the multiset of n "root positions" produced by the
strip-transfer eigenvector (via Chapter 7).
- Q_Z is the Pfaffian from Chapter 3.
- M_Z is the matrix (M(z_i, z_j))_{i, j} from Chapter 3.
- F_Z^{square-point} is a specific Pfaffian normalisation
computable from the "square-point" of the transfer matrix.
At the symmetry point, the matrix in (8.1) is *exactly* the
"crossing determinant" at the "square point" of the transfer matrix.
Concretely, this crossing determinant is
B_n := det(N(x_i, x_j) / H(x_i, x_j))
/ det(Delta(x_i, x_j) / H(x_i, x_j))
at the confluence x_1 = ... = x_n = 1, where:
H(x, y) = x^2 + y^2 + sqrt(2) x y, (8.2)
N(x, y) = H(x, y) + (x - y)(x + y + d), d a fixed real, (8.3)
Delta(x, y) = (x - a) * (Hermite residual of N/H at nodes {b, b, -a}).
Here a, b, d are three fixed real constants from Section 09 of the
source, chosen so that the square symmetry of the transfer matrix
holds. At the square point a = 0, the denominator det(Delta/H)
reduces to the "three-node" Hermite interpolant at nodes {b, b, 0}.
8.4 The three-column Hermite reduction.
Because we evaluate at x_1 = ... = x_n = 1, the denominator
det(Delta(x_i, x_j) / H(x_i, x_j)) is a determinant of a 3-node
Hermite remainder projected onto the degree < n span of polynomials.
Concretely, define for each degree k (0 <= k < n) the "moment"
q_k = q_k(x) := derivative k of V(x) at x = 1,
where V(x) = sin(nu s) / sinh s at x = cosh s. Then:
- The matrix G_{rs} := (2/k) * integral q_r(nu) q_s(nu) w_b(nu) dnu,
r, s = 0, 1, ..., n-1, is positive definite.
- The numerator det(N/H) / (Vandermonde factors) at x_i -> 1 equals
det(G) times the "perturbation" from the N vs Delta numerator.
- The denominator det(Delta/H) / (Vandermonde factors) equals det(G)
times a "three-column correction" from the polynomial interpolant.
The quotient B_n is then:
B_n = det G*_N / det G*_Delta,
where G*_N and G*_Delta are the same as G but with the last three
rows replaced by specific Hermite rows (from interpolating the
coupling N or Delta respectively).
Crucially, the ratio det G*_N / det G*_Delta simplifies to a
determinant of 3 x 3 in the limit, because both numerator and
denominator differ from G in only the same three rows.
8.5 Square-point asymptotics.
From the "polynomial Laplace estimate" of Chapter 6, we know that
the off-diagonal Jacobi coefficients b_j ~ j / (2c) and the monic
orthogonal-polynomial norms are comparable to their "positive 0-weight"
versions. Combined with (8.5)'s structure (Vandermonde-scaled det G),
this gives:
PROPOSITION 8.2 (square-point determinant asymptotics).
B_n as defined above satisfies, as n -> infinity,
B_n = c_B * (1/n) * (1 + O(1/n)) (8.4)
for an explicitly positive constant c_B > 0.
Consequently, after combining with (8.1), (6.13), and the
half-strip normalisation, the strip-crossing mass satisfies
B_H = c_B' * H^{-1/4} * (1 + o(1)), (8.5)
for an explicitly positive constant c_B' > 0.
Proof outline.
(a) The "3-column correction" det(Delta/H) at the homogenised
confluence reduces to a rank-2 or rank-3 object, but the
relevant Hermite interpolation gives
det(three-column finite difference of q_k)
= c_A * n^2 * (leading terms) + O(1),
from the explicit Hermite row structure of Chapter 6.
(b) The "N" numerator has the same n^2 leading order, but with
a *different* coefficient, so B_n becomes a specific *ratio*
of explicit constants. Both constants are nonzero (Chapter
6 positivity of the det G types).
(c) Turning (8.4) into (8.5) requires a continuum limit at
homogeneous z (transfer + eigenvector from Ch 7). The exact
power 1/4 arises from:
c_B * Q_n^2 / R_n * n^{1/4} ~ (J_1) * n^{-3/2} * n^{1/4}
* (Q_n / sqrt(R_n))^2
which becomes H^{-1/4} after using the winding mass formula
J_n * n = sqrt(3)/6 and the exponents of Q_n, R_n from (6.13).
The explicit constant c_B' equals (sqrt(3)/6) times a pure algebraic
factor; we do not reproduce it.
8.6 Consistency check against the winding mass.
At homogeneous z (all z_j = 1) and at the physical parameter, the
asymptotic identity (8.5) can be cross-checked against the winding
mass formula from Chapter 6.
The ratio B_H / B_{2H} should equal 2^{1/4} up to lower-order terms
by (8.5). On the other hand, the winding-mass identity (6.11) gives
n J_n -> sqrt(3)/6 as n -> infinity. Substituting into the
consecutive-identity (3.1.3) at the homogeneous Z, and using the
positivity of Y(Z) at real Z (Chapter 6), gives a consistency check
that the exponents (1/4 in B_H, 1 in J_n * n) match.
Independently, the "rooted polygon mass" of Prop 6.3 evaluated at
H = 1 gives (2 - sqrt2)/3, which agrees with the leading constant
of the H = 1 limit of the "one-port" structure from (6.12) at
homogeneous z. This is a finite check, not a proof of the 1/4
exponent, but it confirms the overall algebraic consistency of the
identity.
8.7 The lower bound.
For the lower bound of Theorem 8.1, we use the same construction in
reverse and positivity. Specifically, we already know from (7.7) and
Prop 7.3 that the total mass in a winding cylinder does not concentrate
on any single loop; on the other hand, the eigenvalue L_n(a) analytic
in a with L_n(0) = B_H * Q_n / R_n (up to constants) has a nonzero
value at a = 0 (by the "positive Gram" of Ch 6 and the identity
(6.12)). Combined with a standard Paley-Zygmund second-moment
argument (which uses the second-moment bound from Chapter 9), this
gives the lower bound B_H >= c H^{-1/4}.
8.8 Summary.
From this chapter we retain the strip-crossing asymptotic
B_H ~ c * H^{-1/4} as H -> infinity, (c > 0). (8.6)
Both directions (c < B_H < C H^{-1/4}) are needed in Chapter 11
(to pin down the height scale of critical bridges).
------------------------------------------------------------------------------
CHAPTER 9. TWO-PORT BOUND IN OBLIQUE SECTIONS
------------------------------------------------------------------------------
9.1 What this chapter proves.
An "oblique section" is a slice of the honeycomb at an angle to a
lattice-side direction: a strip between two parallel lattice-side
lines, but on one part of the slice with "cell ratios" (q, q^2) and
on the other with (q^2, q). Our two-port identity from Chapter 3
gives a specific formula for the mass of polygons that visit two
fixed ports on opposite sides of a slice.
The chapter proves: the mass of such polygons with the two ports
correctly separated is at most C * m^{-4/3} K_0^{1/3}, where m is the
number of unmarked cells between the two marks on the first side, and
K_0 * m is the number on the second. This is the key input to the
second-moment bound of 9.6.
9.2 The two-port identity.
Let m, K_0 be positive integers. Consider an oblique section with:
- m "unmarked" ports between the two marks a, b on the first side;
- K_0 m "unmarked" ports on the second side;
- proportions (1 - theta, theta) of the two cell-ratio types
(1, q) and (q, 1) on each side, 0 <= theta <= 1/2.
Let C_{ab} be the mass of polygons visiting both marks a and b,
where a, b can each be of either rapidity type.
This chapter proves (Prop 9.1):
0 <= C_{ab} <= C * m^{-4/3} K_0^{1/3}, (9.1)
for m >= K_0^{1/4}, with C a constant independent of m, K_0, theta, a, b.
9.3 The algebraic setup.
The two-port mass C_{ab} can be written as a specific sum involving
the half-strip polynomial structure (Chapter 8). At the confluence
of cell-ratio types, it satisfies (Prop 9.2 below): C_{ab} equals
a Pfaffian ratio in the roots of the transfer eigenvector on the slice.
The proof of (9.1) has three steps:
(a) Reduce C_{ab} to a specific determinant (square-point structure
from Ch 8).
(b) Bound the determinant from above by a "discrete Laplace" argument
reminiscent of Ch 8.5 but on the oblique geometry.
(c) Control the constant factor by the ball upper bound.
9.4 The "ball upper bound" comparison.
For later use, we record the "ball upper bound":
sum_{P : diam(P) <= r} X^{|P|} |P|^2 <= C * r^{2 + 1/12}. (9.2)
This is a consequence of the visit comparison (Prop L:eq6):
For a finite convex domain D of triangles with bounded number of
lattice sides and diameter O(r), the sum over chords of critical
mass times length, both endpoints summed, is O(r^{2 + 1/12}).
The proof of the comparison uses:
- The chordal mass through any fixed vertex is bounded by the
vertex transfer (J_1) of Chapter 8.
- The number of vertices of D is O(r^2).
- Total chordal mass = sum over sources of exit mass at each source.
9.5 The oblique bound.
The main estimate (Prop 9.1):
PROPOSITION 9.1 (two-port bound).
With the notation of 9.2, for m >= K_0^{1/4},
0 <= C_{ab} <= C * m^{-4/3} * K_0^{1/3}.
Proof outline.
(a) Determinant reduction. At the "square point" the two-port
mass evaluates as a Pfaffian quotient det(dN/dDelta) structure
as in Ch 8. Explicitly:
C_{ab} = c_A(a,b) * (1 / det G_oblique),
where G_oblique is the "oblique Gram" whose rows are given
by the Hermite derivatives of the oblique eigenvector V(x)
at the two marks.
(b) Determinant bound. The G_oblique matrix has size ~ m on one
side and ~ K_0 * m on the other (two "block" structures).
Its determinant (from the block structure + Hermite)
satisfies
det G_oblique >= c_min * m^{4/3} * K_0^{-1/3}
* (leading positivity constant).
(c) Combine.
The exact constant c_min is computed in Section 09.tex, using the
Christoffel-Darboux identity and the row-vector structure.
We also note that symmetry C_{ab} = C_{ba} reduces to an identity
between the two Gram matrices at acute/obtuse angles (both sides of
the oblique section); this is verified directly at the algebraic
level.
9.6 The second-moment bound.
The second-moment sum is:
PROPOSITION 9.2 (second moment, Prop L:eq10).
For every R >= 2,
sum_{R <= diam(P) < 2R} X^{|P|} * |P|^2
<= C * R^{2/3} * (1 + log log R). (9.3)
Here the sum is over plane polygons modulo lattice translation, and
diam(P) is the Euclidean diameter.
Proof. Decompose the sum over ordered vertex pairs (u, v) of P:
|P|^2 = sum_{(u, v) in P x P} 1.
Fix a "good" ordered pair (u, v) at displacement d(P) >= R^{1/3}
(bounded away from all three lattice-side lines through 0). Each
good pair with displacement d is bounded by at most 1 / (number of
displacements at scale d) times the two-port mass C_d. By (9.1) at
m ~ d, K_0 ~ R/m, this is at most C * d^{-4/3} * (R/d)^{1/3} = C d^{-1} R^{1/3}.
Summing over displacements d from R^{1/3} to R with weight d^{+2}
(the number of P-vertex pairs at distance d), the contribution at
scale d is at most C * d^{2} * d^{-1} * R^{1/3} = C d R^{1/3}.
Summing over d from R^{1/3} to R: sum_{d = R^{1/3}}^{R} d = R^2/2,
times R^{1/3} gives R^{7/3}. This exceeds (9.3)'s R^{2/3} by a
factor R^{5/3}, so the crude vertex-count is too naive. Instead we
use the "band" decomposition of Section 09: on a band of width w
at scale R, the effective number of good pairs is at most c(w/R) |P|^2,
and each costs only C * w * R^{-4/3} after (9.1). Then the
contribution from band w is at most C * R * w * R^{-4/3} = C w R^{-1/3}.
Sum over the O(1 + log log R) width log-bins (from log^2 R to R):
total <= C R^{2/3} (1 + log log R).
See Section 09.tex for the full accounting.
9.7 Summary.
From Chapter 9 we retain:
(i) Prop 9.2 (second-moment bound): sum_{diam <= 2R} X^|P| |P|^2 <= C R^{2/3} polylog.
(ii) Prop 9.1 (two-port bound) for use in Chapter 10 (irreducible
bridge diameter tail).
------------------------------------------------------------------------------
CHAPTER 10. BRIDGES, CAPS, ENCLOSING LOOPS, GROWTH
------------------------------------------------------------------------------
10.1 Bridge geometry: definition and tightness.
Recall (Ch 1, 2, 8):
- A bridge of height H >= 1 joins a fixed midport on the bottom of
a horizontal-strip (level 0) to any midport on the top (level H),
visiting only centres strictly between the two lines.
- An arch of gap g >= 1 is a port-path from a fixed port on the
boundary of a half-plane to the port g units away, meeting the
boundary only at its own two ends.
For the bridge, the "transverse position" of the top port is on the
infinite horizontal line. The following lemma controls it.
Lemma 10.1 (tightness).
Fix T > 0. There is a constant c = c(T) > 0, independent of H,
such that the total mass of bridges of height H whose entire path
stays within lateral displacement T*H of the starting port, with
any specified weak side (left or right) of the target, is at
least c * H^{-1/4}.
Equivalently, bridges are not horizontally "unbounded": at least a
positive fraction of B_H remains within a linear cone.
Proof sketch (Section 06.tex "tightness"). Clip the strip by two
lateral lattice-side lines at distance ~ S >> H; the beyond-cone
mass equals (via the flux identity on the half-plane) the mass of
arches of diameter > S, which is <= C S^{-1/4}. Choose S of order
H, so the tail is a fixed small fraction. Reflection symmetry gives
either side.
10.2 The cap lemma.
The cap lemma is the boundary analog of B_H.
PROPOSITION 10.2 (caps at prescribed gaps).
There are constants c, C > 0 such that for every integer g >= 1,
sum_{omega an arch at gap g, diam(omega) <= Cg} mu(omega)
>= c * g^{-5/4}. (10.1)
Moreover the unrestricted arch masses satisfy
A_g >= c * A_q whenever 1 <= g < q,
for another constant c > 0 (same proposition, weaker form).
The proof (Section 10.tex "gap comparison") uses:
(a) For each sufficiently large r there is an integer g' in
[floor(r), Cr] with A_{g'} >= c r^{-5/4} and diameter <= Cr.
This comes from averaging over lateral displacements of a
fixed source across the boundary.
(b) "Gap comparison": add a one-cell-thick bump of rhombi below
the boundary between ports g and q and use the half-plane flux
identity to compare A_g(D) with A_q(D') for enlarged domains
D' subset D. The comparison gives A_g >= c A_q for 1 <= g < q.
(c) Pointwise formula. For large g, a sum over integers j in
[g/2, g) of A_j >= c A_g * g (from (b)) combined with the
diameter tail sum_{j in [g/2,g)} A_j = O(g^{-1/4}) (which comes
from the diameter tail Prop L:root-ed6 in Section 10) gives
A_g = O(g^{-5/4}). Combined with (a):
c g^{-5/4} <= A_g <= C g^{-5/4}.
(d) Diameter control. The added "bump" is O(g) wide, so the
retained arches have diameter <= C' g for a fixed C'.
The exponent 5/4 is exactly the boundary-weight prediction
1 - 2 * (5/8) = -5/4.
10.3 Finite chord mass (Prop box-moment).
The "enclosing-loop" finite-mass bound already appeared in Ch 9.4
for the 2+1/12 exponent. Here we need a sharper version, on a
length-weighted scale.
Recall (Ch 1) the "visit comparison": for a finite convex polygonal
domain of triangles of diameter at most C_0 r, of bounded lattice
complexity, the critical mass of chordal arcs with both ends free
is at most C r^{2+1/12}. Then:
PROPOSITION 10.3 (finite chord length mass, Prop box-moment).
For every finite convex domain of triangles with a bounded number
of lattice sides and diameter at most C_0 r, the critical
length-weighted mass of all boundary-to-boundary chords, with both
endpoints summed, is at most C r^{25/12}.
In particular, for a fixed T > 0, the length-weighted mass of
strict bridges of height d from one fixed source, whose lateral
displacement stays within T d, is at most C_T d^{13/12}.
Proof: By the ball upper bound of Ch 9.4 and (visit comparison
revisited). Centers right at the boundary number O(r) and each
contributes at most O(r) (since the exit mass from any source is
bounded), so total is O(r^2) beyond the arc-mass bound; the leading
contribution is the interior arcs. The transverse-displacement
version is a translation argument over O(d) distinct sources in a
common enlarged box.
10.4 Recoverable half-plane growth.
The main geometric construction of the entire paper: from a fixed
port on a boundary, we can find a large family of "prefixes" whose
lengths are controlled and whose "cut" is selected by a deterministic
rule. This is the input to Ch 11's bridge construction.
Set-up. Let n, u be level coordinates in two lattice directions
whose unit inward vectors have dot product 1/2. Let A be a port on
the n = 0 wall.
The "first qualifying cut" for a forward path starting at A is the
lowest candidate u-level at which the path crosses exactly one port,
whose prefix alone satisfies:
- diameter at most C_fc r (a fixed bound);
- confinement in n > 0;
- terminal n-coordinate at least c_fc' r;
- optional length at most M r^{4/3} (a fixed constant M).
PROPOSITION 10.4 (recoverable half-plane growth, Prop recoverable-growth
+ Prop length-controlled-growth).
There are constants r_0, c, C, c' > 0 with the following property.
For every r >= r_0 there is a family F of distinct prefixes from
A such that:
(i) total critical mass of F is at least c;
(ii) each prefix lies in u < s for some s in a fixed u-bin
[r, 2r];
(iii) every prefix has all centres in n > 0, diameter at most
C r, length (number of centres) at most C r^{4/3};
(iv) every prefix's terminal n-coordinate is at least c' r;
(v) at the terminal u-bin, the cut is the first-qualifying cut
and gets reproduced exactly under any extension of the path
strictly beyond the cut.
Proof outline (Section 09.tex). The half-plane identity gives,
for each source, a total mass of "boundary chords" of the ball at
radius r that is comparable to a constant (independent of r, in the
visit-comparison normalisation). Cut off chords of length > M r^{4/3}
at a cost of a fraction (by Prop 10.3); cut off chords that escape
the T-narrow band at a cost of a fraction (by tightness Lemma 10.1);
cut off chords that fall outside the n > c' r zone at a negligible
cost. The remainder is a family that satisfies (i)-(iv). Select
a "cut" via the first-qualifying rule; by the bracket of the first-cut
criterion, extensions strictly beyond the cut preserve the cut.
The point of (iii)-(iv) is that length ~ r^{4/3} and vertical
displacement ~ r are in a definite ratio, exactly the
SAW length/diameter scaling.
The dependence of length on diameter: r^{4/3} = r * r^{1/3}, which
is where the exponent 4/3 = 1 + 1/3 appears.
10.5 Critical irreducible bridges.
As a downstream standard consequence (not needed for Ch 11) we
record the classical decomposition of bridges into irreducible
bridges.
The decomposition. Cutting a strict bridge at all its intermediate
"single-crossing" levels decomposes it uniquely into "irreducible"
bridges with no intermediate single-crossing level. Cuts are
ordered by height. Each irreducible portion lies strictly between
its two cut lines. Conversely, irreducible pieces concatenate
freely (disjoint slabs, self-avoiding by disjointness, weights
multiply).
PROPOSITION 10.5 (critical irreducible law).
The sum of X^{|gamma|} over irreducible strict bridges gamma from
a fixed port is one. Under this probability law, if Y and D are
the height and diameter of the sampled bridge, then
1 - E exp(-tY) ~ t^{3/4} (0 < t <= 1),
Pr(D > r) <= C r^{-3/4} (r >= 1).
Proof outline (Section 10.tex). Let i_j be the total irreducible
mass at height j >= 1. Unique factorisation gives
sum_{h >= 0} B_h z^h = sum_{k >= 0} ( sum_{j >= 1} i_j z^j )^k.
At z = e^{-t} < 1, the left side is finite (B_h ~ h^{-1/4}), so
the right side converges for I(t) = sum i_j e^{-tj} < 1. Then
1 - I(t) = ( sum_{h >= 0} B_h e^{-th} )^{-1} ~ t^{3/4}.
The last follows from B_h ~ h^{-1/4} by comparison to the integral.
Monotone convergence as t -> 0 shows sum i_j = 1 (mass = 1) and
gives the height tail.
For diameter: fix the transverse-confinement constant T from
Lemma 10.1; pick c_0 small so a bridge of height ~ c_0 r
confined in T r has diameter <= r. The total mass of such confined
bridges at height in [c_0 r, 2c_0 r] is at least c r^{3/4} (from
Lemma 10.1 and the H^{-1/4} scaling summed over c_0 r heights).
Accounting for irreducible-piece diameter <= r (each piece fits
in the original bridge), we get
c r^{3/4} <= sum_{k >= 0} a_r^k, a_r = Pr(D <= r).
If a_r = 1 the tail is immediate; else sum = 1/(1-a_r), giving
1 - Pr(D <= r) <= C r^{-3/4}.
10.6 What we retain.
(i) Prop 10.2 (cap masses A_g ~ g^{-5/4}, all-orders tail).
(ii) Prop 10.3 (finite chord length mass, for length control).
(iii) Prop 10.4 (recoverable growth, core geometric input for Ch 11).
(iv) Lemma 10.1 (tightness of transverse displacement).
(v) Prop 10.5 (irreducible bridges; consequences only used as a
consistency check in Ch 11).
------------------------------------------------------------------------------
CHAPTER 11. FROM MOMENTS TO THE LOGARITHMIC LENGTH WINDOW
------------------------------------------------------------------------------
11.1 The theorem, restated.
THEOREM (Theorem A, restated with details).
There exist constants c, C > 0 and h_0 > 0 such that for every
real h >= h_0, setting
s(h) = h * (log h)^{1/128},
R(h) = h * (log h)^{1/64},
ell_-(h) = h^{4/3} * (log h)^{-1/8},
ell_+(h) = 2 h^{4/3} * (log h)^{1/2},
we have
sum_{1 <= H <= 2 R(h)}
sum_{omega in Bridges(H), ell_-(h) <= |omega| <= ell_+(h)}
mu(omega) >= c * h^{3/4} * (log h)^{-C}.
(Counting conventions as in Ch 2.)
This chapter proves the theorem. The proof has three stages:
(a) Retain paths of moderate diameter and moderate length.
(b) If they are already bridges, we are done.
(c) Otherwise they are arches; attach two disjoint exterior
"connectors" to convert them to bridges, retaining a definite
fraction of the mass.
11.2 The length/diameter cutoffs.
We apply Prop 10.4 (recoverable half-plane growth) with r = h. It
gives a family of prefixes with total mass >= c_h (a fixed positive
constant) from any fixed boundary port, all with length <= C h^{4/3}
and terminal n-coordinate >= c' h.
We want to *also* cut off long diameters and to select a specific
length window.
Retain first-moment budget. From B_H ~ H^{-1/4} (Theorem 8.1),
the total mass at height scale ~ h is c h^{-1/4}. We want to keep
a definite fraction of it inside:
- diameter at most s(h) = h (log h)^{1/128};
- length in [ell_-(h), ell_+(h)] = [h^{4/3}(log h)^{-1/8}, 2 h^{4/3}(log h)^{1/2}].
The diameter cutoff is what makes the "s(h)" appear. The length
cutoff determines the "log h" tuning.
LEMMA 11.1 (retained mass).
For every sufficiently large h, the critical mass of bridges of
height between h and 2h whose entire path has diameter at most
s(h) and length in [ell_-(h), ell_+(h)] is at least
c * h^{-1/4} * (log h)^{-7/12}, for a constant c > 0.
Proof. From Theorem 8.1, B_H >= c H^{-1/4} for each H in [h, 2h].
The total mass in the strip of height ~h is thus >= c h^{3/4}
(the sum of ~h heights, each ~h^{-1/4}). Cut off long diameters at
scale s(h): the mass with diameter > s(h) is at most C s(h)^{-1/4}
(by the arch diameter tail of Prop 10.2 in the tightness Lemma
10.1). At height scale h, this tail is
C s(h)^{-1/4} = C h^{-1/4} * (log h)^{-1/512},
which is a fraction of the total.
Cut off short and long lengths. By part (iii) of Prop 10.4, the
retained prefixes have length at most C h^{4/3}. We further restrict
attention to the length window
ell_-(h) <= |omega| <= ell_+(h),
which concentrates on the natural length scale h^{4/3} up to log
factors. Mass outside the window is a lower-order polylog contribution
by Prop 10.3 (finite chord mass at scale s(h)): the length-weighted
mass at scale s(h) is at most C s(h)^{13/12} = C h^{13/12} polylog,
and dividing by the lower threshold ell_-(h) ~ h^{4/3} polylog gives
mass >= c h^{-1/4} polylog retained inside the window. Together with
the diameter cutoff this yields the retained-mass lower bound of
Lemma 11.1.
Combining the cuts, the retained fraction is at least
c * h^{-1/4} * (log h)^{-7/12}
for a fixed c > 0. This is the general form used below.
11.3 The two cases: bridges or arches.
Split the strip of height h into bridges (top port at level H) and
arches (both ports on level 0 or level H).
If the total B_H mass in the strip is at least (1/2) c h^{3/4}, we
are done: choose H = 2h in the theorem.
Otherwise, at least half the mass has its retained paths living as
arches. Fix one wall (top or bottom) and one gap direction
(left/right) as the "primary" wall: reflection and reversal reduce
to one wall and one direction at a fixed-factor cost.
The retained arches have:
- gap >= h / log h,
- length in [ell_-(h), ell_+(h)],
- diameter at most s(h),
- total mass (of the sub-family) at least c h^{-1/4} (log h)^{-K}
for a fixed K.
The trick (below) converts them to bridges by attaching connectors.
11.4 The exterior connectors.
Set R := the target bridge height, in the range [R_0(h), 2 R_0(h)]
where R_0(h) = h (log h)^{1/64}. Fix one integer R in this range.
For each retained arch with right endpoint B and gap g, construct a
bridge by:
- Positioning the arch above the line y = 0 with left endpoint at
a fixed port A.
- Attaching, below y = 0, two connectors:
Left connector: a forward path from A that escapes to the
"far left" and turns to reach level R on the opposite side.
Right connector: a forward path from B that escapes to the
"far right" and turns to reach level R.
- The connectors together with the arch form a bridge of height
R (the far-left and far-right turns make the top ports live at
level R, on the appropriate side of the strip).
The geometry ensures connectors can be made disjoint from each other
and from the arch by choosing the "cut" of Prop 10.4 to be the
arch's far endpoint.
Lengths. If the arch's length is in [ell_-(h), ell_+(h)], and each
connector has length ~ R^{4/3} * (log h)^{-some small power}, then
the total bridge length is:
|omega_total| ~ |arch| + |connector_L| + |connector_R|
<= ell_+(h) + 2 * R_0(h)^{4/3}
<= ell_+(h) + 2 * h^{4/3} (log h)^{4/192}
~ h^{4/3}(log h)^{1/2} + O(h^{4/3} polylog).
Since R_0(h)^{4/3} = h^{4/3} (log h)^{1/48}, but our ell_+(h)
targets h^{4/3} (log h)^{1/2}, we have extra room. Similarly,
the lower bound |omega_total| >= ell_-(h) is satisfied as long as
the connectors don't shrink the total below.
The point is: connectors of length ~ R_0(h)^{4/3} fit inside the
length window.
Mass of connectors. Each connector has mass comparable to R_0(h)^{-1/4}
from Prop 10.4. So a bridge assembled from one arch + two connectors
has mass <= (arch mass) * (connector mass)^2 / (normalization).
11.5 Recoverability.
The converse direction is the crucial part: given a bridge of
length in [ell_-, ell_+] and diameter <= s(h), how many (arch,
connector_L, connector_R) decompositions can it have?
The construction of Prop 10.4 includes a *recovery rule*: the cut
is the first-qualifying cut. So from the bridge geometry alone,
the two "cut walls" (below the level-0 line) are identifiable by the
first-qualifying cut applied to the bridge's left and right
endpoints.
Once the two cuts are recovered, the arch's surface is exactly the
bridge above level 0. The connectors are recovered as the paths
strictly below the cuts, since they end at the bridge's endpoints.
MULTIPLICITY. The recovery is not 1-to-1: each bridge could
arise from potentially many choices of (arch, connectors). The
"first qualifying cut" rule is what makes the multiplicity small:
- Once the bridge is given, its left and right cut levels are
determined as the first u-levels on each side at which the
bridge prefix (below that level) satisfies the criteria of
Prop 10.4. This is unique (there is exactly one lowest such
level per side) by construction.
- The arch above the cut is then determined as the top part of
the bridge above level 0.
- The two connectors are the parts strictly below the two cut
levels; their starting ports are the two cut ports, and their
endpoints are the two bridge endpoints.
- The remaining freedom is only the choice of "labels" on each
connector -- i.e. the specific levels used in the first-cut
rule. These labels range over integers in a fixed polylog
window of size (log h)^{C_0}.
Hence M(bridge) = O((log h)^{C_0}) for a fixed C_0 > 0. The
multiplicity is polylog in h, NOT polynomial. This is the crucial
fact that drives the theorem: without the first-cut rule, the
"obvious" multiplicities would be R^{O(1)} and the theorem would
lose powers of h, not just logs.
11.6 Closure: bridge mass at a single height scale.
Let R be a fixed height scale in [R_0(h), 2 R_0(h)], where
R_0(h) = h (log h)^{1/64}.
Let F_A be the total mass of retained arches (diameter <= s(h),
length in [ell_-, ell_+], gap >= h / log h) at that scale. By
Lemma 11.1, F_A >= c h^{-1/4} (log h)^{-7/12}.
Let F_B be the total mass of bridges of height in [R, 2R] with
length in [ell_-, ell_+] and diameter <= s(h).
CLAIM. F_B >= F_A / M, where M = O((log h)^{C_0}) from 11.5.
Proof of claim. The map (arch) -> (bridge) from 11.4 is well-defined
and injective on retained arches at fixed anchor A: different arches
give different bridges above level 0. Conversely, each bridge of the
F_B type decomposes into at most M (arch, connector) data by 11.5.
Therefore
F_A <= sum over bridges in F_B of (contributing arches per bridge)
<= M * F_B.
Thus
F_B >= F_A / M >= c h^{-1/4} (log h)^{-7/12 - C_0}. (11.1)
11.7 Finishing: summing over heights.
Now combine with the strip-crossing bound B_H ~ H^{-1/4} of
Theorem 8.1.
Fix a dyadic height scale k and apply (11.1) at h = k. The
retained-arch mass at that scale is c k^{-1/4} (log k)^{-7/12 - C_0}.
The number of integer heights H in a dyadic band [k, 2k] is k.
Summing over dyadic k from 1 to 2 R_0(h) (the theorem's height
range) gives
sum_{H=1}^{2 R_0(h)} (retained bridges at height H)
>= c (log 2R_0(h))^{-7/12 - C_0} * sum_{k=1}^{2 R_0(h)} k^{-1/4}
~ c' R_0(h)^{3/4} (log R_0(h))^{-7/12 - C_0}
= c' h^{3/4} (log h)^{3/256 - 7/12 - C_0}.
Setting C = 7/12 + C_0 - 3/256, a fixed positive constant (since C_0
is a fixed polylog exponent), we get
sum_{H=1}^{2 R_0(h)} sum_{ell_-(h) <= |omega| <= ell_+(h)}
mu(omega) >= c' h^{3/4} (log h)^{-C}.
This is Theorem A.
11.7.1 Why the two-scale structure works.
The apparent "polylog loss on one side, polylog gain on the other"
is the crux of the argument. We lose (log h)^{-C_0} to the recovery
multiplicity (11.5) and (log h)^{-7/12} to the length-window
restriction (11.2); we gain (log h)^{1/64} to the height scale
R_0(h) and (log h)^{1/(3*64)} to the integrated height sum. The
net loss is polylog in h and the total height gain is a fixed power
of h; both fit inside the fixed constant C > 0.
11.8 What remains open.
The theorem gives mass inside a logarithmic window
[h^{4/3} (log h)^{-1/8}, 2h^{4/3} (log h)^{1/2}] summed over heights
in [1, 2h (log h)^{1/64}]. It does NOT prove:
- The mean length of a critical crossing is asymptotic to a
constant times h^{4/3}. The mean could in principle be
dominated by rare very long crossings.
- Concentration at a specific height, e.g. H = h.
- Concentration at a specific length, e.g. |omega| = h^{4/3}.
- Precise constants for the asymptotic B_H / H^{-1/4}.
- Conformal invariance / SLE_8/3 for the scaling limit.
These are independent conjectures (partly on Conformal Loop
Ensembles), established neither here nor in the cited references.
What the theorem does provide: an unconditional, quantitative lower
bound on critical mass inside a natural, physics-motivated length
window, with only a logarithmic loss. That is enough to
exclude "all critical mass concentrates at a single pathological
configuration" -- the main trivializing scenario.