SIGNED CYLINDER PROPAGATION AND MARKED POLYGONS
ON THE HONEYCOMB LATTICE
A plain-language edition, written to be read front to back.
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READ ME FIRST
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This paper is about self-avoiding walks and polygons on one
specific graph, the honeycomb lattice, at the exact density
where they sit on a knife edge -- neither dying out nor
blowing up. It studies three observables: a cylinder
partition with two marked points, a finite chord law inside
a regular hexagon, and a squared-length sum for planar
polygons. Each has a clean power-law growth, proved here.
It is written as one continuous story. Every technical word
is defined on first use, every theorem is stated with its
exact exponents, and every proof step is explained in words
before its algebra. If you meet a symbol you do not know,
jump to the GLOSSARY at the end.
The whole paper in one paragraph:
Draw self-avoiding walks and polygons on the honeycomb
lattice, weighting a length-N walk or polygon by x^N, where
x = 1/sqrt(2+sqrt(2)) is the honeycomb critical weight.
Consider families of disjoint polygons: sometimes wrapped
around a cylinder and required to separate two marked
points; sometimes constrained to lie inside a big hexagon
and to surround its central face; sometimes pinned to
pass through two specified bonds on a tilted cylinder.
Each of these sums grows like a clean power of the size
parameter as it goes to infinity. On the balanced cylinder
at fugacity g = 2 the partition grows like m^{1/6}. Its
curvature at zero fugacity grows only logarithmically.
The planar hexagon variant grows like R^{1/12}. The chord
law inside the hexagon has mean length R^{4/3} and uniform
bulk vertex visit probability R^{-2/3}. The length-squared
mass of planar polygons of diameter at most H grows like
H^{2/3}. The engine is a signed transfer matrix
representation whose cancellations are controlled by
absolute estimates strong enough to still yield provable
power laws even though the individual terms are signed
or complex.
A note on signs. Every physical weight here is positive.
But the standard exact transfer representation of these
partitions uses signed or complex "auxiliary" weights, in the
spirit of the Duminil-Copin-Smirnov local cancellation
theorem. To make the signed representation rigorous, one
must show that the absolute values of the signed weights are
themselves summable, uniformly in size. That absolute
summability step is the technical heart of this paper; only
after it does the exponent extraction become an honest
inequality on the physical partition rather than a formal
manipulation.
How to read this edition:
PART 1 fixes the graph, walks, polygons, and observables.
PART 2 states all theorems with exact exponents.
PART 3 is the one-page map of how the proofs go.
PART 4 proves the cylinder theorem (Theorem 1).
PART 5 proves the chord and visit corollaries and the
length-squared estimate.
PART 6 proves the two-bond marked interval estimate that
powers PART 5.
PART 7 says what is new relative to earlier work.
PART 8 is honest about what is not claimed.
PART 9 is the glossary.
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PART 1. THE STAGE
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1.1 The honeycomb lattice.
Start with the regular infinite triangular tessellation of
the plane, of side length one. Every triangle -- pointing up
or down -- has a center. Join the centers of any two
triangles that share a side. The result is the honeycomb
graph: three edges meet at every vertex, and the graph looks
like an infinite sheet of chicken wire.
Two length measurements are used below. Euclidean distance
is the usual distance in the plane. The diameter of a
bounded set is the largest Euclidean distance between two
points of the set.
1.2 Self-avoiding walks and polygons.
A walk in a graph is a sequence of vertices where successive
vertices are adjacent. It is self-avoiding if no vertex is
visited twice. Its length is its number of vertices. A
polygon is a closed self-avoiding walk; its length equals
its number of edges.
Let x be the critical weight
x = 1 / sqrt(2 + sqrt(2)) = 0.5411961... .
This is the reciprocal of the honeycomb connective constant,
a classical constant whose exact value was established by
Duminil-Copin and Smirnov (2012). At criticality, walks
weighted by x^N sit exactly at the phase transition between
short and long typical lengths.
A walk of length N contributes x^N to every sum below. A
polygon of length |lambda| contributes x^{|lambda|}. Two
polygons are "incompatible" if they share at least one
vertex; a polygon is incompatible with itself. Weights
multiply: an incompatible pair has weight x^{|lambda|+|mu|}.
1.3 Ports and first-exit walks.
For a region D in the plane (a hexagon, a half-plane, a
convex cap), a "port" is the midpoint of a tessellation
side on the boundary of D. A "first-exit walk" from a
starting port is a self-avoiding walk inside D that never
visits a boundary port except at the start and at its very
last vertex, which is a boundary port other than the start.
First-exit walks are important because the regional
partition of these walks satisfies an exact signed balance:
the Duminil-Copin-Smirnov cancellation. This balance makes
the total first-exit weight from any fixed port to any
convex boundary comparable to one. Section 5.1 turns that
balance into the quantitative arch estimate that controls
long walks.
1.4 Cylinders and separating cycles.
Take a horizontal cut of the honeycomb with 2m sites
(m a positive even integer through the paper) and identify
sites modulo 2m: that is, wrap the strip into a horizontal
cylinder. On the cut, mark two points O = (0,0) and
O' = (m,0); they divide the 2m sites into two equal lists
A and B of size m.
A "cycle" on this cylinder is a simple closed curve made of
honeycomb edges, possibly wrapping nontrivially around the
cylinder. It is "separating" if O and O' lie in different
connected components of its complement. A "family" is a
collection of pairwise disjoint separating cycles (they
share no vertex).
1.5 The cylinder partition G_m(g).
Define G_m(g) as the sum, over families of pairwise disjoint
separating cycles, of
g^k * x^{|lambda_1| + |lambda_2| + ... + |lambda_k|}
where k is the number of cycles in the family, and
|lambda_i| denotes the polygon length. The empty family
contributes 1. Because any separating cycle crosses the
cut between the two marks at least once, and there are
only 2m sites available, G_m(g) is a polynomial in g of
degree at most 2m.
The cylinder is infinite in the vertical direction. The
definition is by a limit: take finite cylinders with both
rims closed (meaning no boundary port is used), let the
height tend to infinity, and take the limit of partitions.
Section 4.1 proves this limit exists.
1.6 The hexagon and the chord law.
Let D_R be a regular hexagonal patch of honeycomb of side
length R (R a positive integer). Let Z(R) be the sum, over
collections of pairwise disjoint polygons contained in D_R
that surround its central face, of
2^k * x^{|lambda_1| + ... + |lambda_k|},
where k is the number of polygons. The factor 2 per polygon
is the "fugacity 2" convention.
Let C_R be the set of first-exit chords of D_R of diameter
at least R/10. Here a "chord" is a first-exit walk whose
two endpoints are boundary ports; the total sum runs over
every starting port and every possible first-exit port, and
retains the direction from start to exit. Define the finite
partition and its probability law by
Z_R^chord = sum_{gamma in C_R} x^{N(gamma)},
P_R^chord(gamma) = x^{N(gamma)} / Z_R^chord.
The chord law is a finite, normalized probability measure on
the chords of the hexagon. Theorem 2 gives its total mass,
its mean length, and its bulk vertex visit probability.
1.7 Planar polygon sums.
For the squared-length estimate, sum over planar polygons
modulo lattice translations (two translates count as one
class):
S(H) = sum_{diam(lambda) <= H} x^{|lambda|} |lambda|^2.
The weight |lambda|^2 is the number of ordered pairs of bonds
inside lambda. Summing over such pairs turns S(H) into a sum
over two-bond configurations
(a polygon containing both bonds at once)
weighted by the polygon's x^weight. That reformulated
two-bond sum is the positive quantity that Theorem 3
estimates, and Theorem 3's two-bond estimate is what
produces the H^{2/3} bound on S(H).
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PART 2. THE THREE THEOREMS
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First, a notational convention that appears in every theorem.
"F(R) = R^{a+o(1)}" means: log F(R) / log R -> a as R grows.
An upper bound "F(R) <= R^{a+o(1)}" means: for every epsilon > 0
there is a constant C_epsilon such that
F(R) <= C_epsilon * R^{a+epsilon} for all sufficiently large R.
A lower bound "F(R) >= R^{a+o(1)}" is analogous.
A statement is "uniform" in some data if the constants and the
thresholds do not depend on that data. Every theorem below
is uniform in the sense that its constants and thresholds may
depend on the fixed parameters (which are named in the
theorem) but not on the growing parameters m, n, R, or H.
Now the three theorems.
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THEOREM 1 (Cylinder pressure and planar nesting).
As m grows through even positive integers, the balanced
cylinder partition G_m(g) with two marks satisfies:
(1a) G_m(2) = m^{1/6+o(1)}.
(1b) - (d^2/dg^2) log G_m(g) at g = 0 is O(log m).
Here the derivative counts the total weight of
ordered pairs of cycles that are incompatible --
i.e., pairs of separating cycles that share a
vertex, together with each cycle paired with itself.
(1c) Locally uniformly for complex Theta with
|Re Theta| < pi,
|G_m(2 cos Theta)|
<= m^{Re(1/6 - 2Theta^2/(3 pi^2)) + o(1)}.
(1d) The planar nesting partition Z(R) of a hexagon of
side R surrounding its central face satisfies
Z(R) = R^{1/12+o(1)} as R -> infinity through
integers.
The informal reading:
(1a) says a balanced cylinder with two marks has a
partition at fugacity 2 growing only like the sixth
root of its circumference. Cycle families do not fill
up the cylinder; almost all of the mass is a short
burst around each mark.
(1b) says the "curvature" of the logarithm of the
partition at zero fugacity grows logarithmically. This
is a strong statement: the total mass of "bad" pairs
of cycles (those that share at least one vertex) is
small. It is what makes the localization argument
(Section 5.4) possible.
(1c) is the same as (1a) at Theta = 0, extended to
complex fugacity. It is what allows contour moves
and three-circle inequalities later.
(1d) is the planar limit. Set fugacity 2, so a family
of k polygons gets weight 2^k. The planar nesting
partition grows like the twelfth root of the size.
Combining (1d) with the chord-partition estimate
yields the R^{4/3} chord mean below.
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THEOREM 2 (Finite chords and marked polygons).
For the regular hexagon of side R and its chord law with
diameter at least R/10 (as defined in Section 1.6):
(2a) Z_R^chord = R^{3/4+o(1)}. The total chord
partition, summed over every start and every exit,
grows like the fourth root of R per eighth power.
(2b) E_R^chord N = R^{4/3+o(1)}. The mean chord length
under the chord law grows like the four-thirds
power of R. This matches the standard 2D
self-avoiding walk exponent 1/nu = 4/3, with the
precise normalization and boundary conditions
fixed in Section 1.6.
(2c) Uniformly for honeycomb vertices v at distance
at most R/10 from the hexagon's central face,
P_R^chord(v lies on the chord) = R^{-2/3+o(1)}.
Every bulk vertex is visited with the same
power-law probability.
(2d) For planar polygons summed modulo lattice
translations,
sum_{diam(lambda) <= H} x^{|lambda|} |lambda|^2
<= H^{2/3+o(1)}.
The informal reading:
(2a) says the total chord mass grows like the fourth
root of the size parameter. The boundary region
contributes a critical amount that is comparable to
what would appear in the full plane, up to a factor of
about R to the fourth root.
(2b) says the normalized mean chord length is exactly
R^{4/3}, matching the Flory-type prediction for 2D
self-avoiding walks. The proof of (2b) uses (2a) plus
(1d): the mean is a ratio of a sum over vertices to
the total mass. This is Proposition 5.3 in Part 5.
(2c) says that the chord partition is "delocalized":
no bulk vertex is meaningfully more visited than any
other. This is what one expects at criticality.
(2d) bounds the second moment of the side-length of a
polygon by its diameter. The power 2/3 is exactly
2/nu - 2 with nu = 3/4; the difference between
|lambda| = N vertices and diam ~ N^{3/4} is absorbed.
Theorem 2(d) is a positive-moment estimate for the
finite diameter truncation.
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THEOREM 3 (Unequal marked intervals).
Take a "physical tilted cylinder": a periodic cut of the
honeycomb whose steps are drawn from the trigonometric
lattice {1, tau} with tau = e^{i pi/3}, made from two
horizontal cuts joined by a shifted cut, so that both
marked bonds cross cut steps parallel to one another
(called "horizontal-type").
Its two cut intervals have m+1 and n+1 sites, with
n >= m and M = m+n+2 even. In each unmarked list the
logarithmic site coordinates have imaginary parts -f
and 1-f, where 0 <= f <= 0.6. Their respective counts
differ from m(1-f), mf in the first unmarked list and
from n(1-f), nf in the second by at most a fixed
constant.
Let H^poly be the total critical weight of a single
unoriented polygon that passes through both marked bonds:
H^poly = sum over (single simple unoriented cycle
through both marked bonds) of
x^{length of cycle}.
This sum is finite (the same walk cannot wind around
a period of length > 2m too many times without
self-intersecting).
THEOREM 3 (restated). Uniformly over these lists and
their orders, for every epsilon > 0 there are C_eps and
m_eps such that
H^poly <= C_eps * n^eps * m^{-4/3} * (n/m)^{1/3}
for every n >= m >= m_eps.
The informal reading:
Theorem 3 is the two-bond estimate that powers the
length-squared result. If you pick any two bonds at
distances of order m and order n, and sum the weights
of all polygons passing through both, the total
falls off like m^{-4/3}, then grows like the cube
root of the aspect ratio n/m. The 4/3 is the
polygonal length exponent visible in (2b); the
1/3 = 1 - 2/3 is the balancing exponent that makes
the sum over pairs of bonds (Section 5.7) converge
to the H^{2/3} bound of Theorem 2(d).
Theorem 3 is proved independently of Theorem 1 and
Theorem 2, in Part 6. Its proof uses the same signed
transfer representation but for two marked bonds
instead of one unmarked cycle.
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A note on how (1a), (2a), (2b), (2d) fit together.
Theorem 1 gives the balanced cylinder at fugacity 2.
Theorem 5.4 (localization) drops every cycle that
travels far from the marks, giving Z(R) = R^{1/12}
for the planar case. Proposition 5.3 then compares
the chord mass to the visit mass, giving (2a),
(2b), (2c). Theorem 3 supplies a separate positive
mass estimate; Parts 5.6 and 5.7 turn that estimate
into (2d).
So the paper proves one main theorem (Theorem 1),
two corollaries (Theorems 2a, 2b, 2c), and one
largely independent two-bond estimate (Theorem 3)
whose only downstream use is the length-squared
bound (Theorem 2d). This is why Part 5 does two
things: prove the chord corollaries, and separately
produce (2d) using Theorem 3.
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PART 3. THE MAP OF THE PROOF
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Here is a one-page map before the details. Each numbered step
becomes a section in Parts 4-6.
3.1 Theorem 1 -- the cylinder exponent.
Read this list twice; every term is defined later.
Step 1. Orient every occupied edge of the honeycomb in one of
three states: empty, forward along the edge, or backward.
Attach a phase x^{1/3} times a 1/8-th of pi phase to each
local turn, a sign to each seam crossing, and a phase to each
crossing of the "A cut". Then, over all self-avoiding
configurations inside a finite closed cylinder, the oriented
sum vanishes unless every cycle separates the two marks;
a separating cycle contributes g. Section 4.1.
Step 2. Let the cylinder height grow. The resulting "vacuum
pairing" converges, because the total mass of half-plane
arches is finite (Section 5.1). You get two vectors indexed
by spin configurations on the cut. Section 4.1.
Step 3. The local Yang-Baxter equation and two "fusion"
identities (both proved in Section 4.2) let you separate
the two vacuum vectors into an exact finite sum over a
single three-valued spin variable per site. This is the
"finite spin sum" of Section 4.3.
Step 4. Individual terms of this finite sum have poles when
two site coordinates coincide. The combined sum is
regular -- the poles cancel. Section 4.4 replaces the
finite sum by a contour integral (with each site
represented by a residue) and then by a fermionic trace on
an angular space of circumference T. Section 4.5 (the
"bosonization") converts this into a horizontal gas on a
circle of auxiliary length T, with two species of particle
carrying integer charges.
Step 5. The gas has signed and complex weights. Section 4.6
proves an absolute stability bound: the total absolute
weight of any configuration of gas particles is at most
exp(-c * sum of |particle count per box|^2 - c * |running
charge at box boundaries|^2). This bound is what lets you
sum the signed series absolutely, and is the deep technical
input.
Step 6. The gas can be rewritten as a compact block
propagator with an extra contracting coordinate that keeps
the "tail" of the interaction exactly. Section 4.7
constructs this operator. Its empty-contour power traces
identify E_0 (all types) with spectral radius r_E = e^{-b
pi / 16}, and I_0 (only the unsuppressed types) with
eigenvalue 1.
Step 7. The site factors omega_tau(m, x) that multiply each
particle have modulus < 1 off the origin and approach 1 at
infinity, with an exponential correction of rate pi/3.
Therefore on an interval of length L ~ log m around the
origin, all suppressed types disappear. Outside this
interval, the exterior operator still allows all types. The
inner rate is r_I = 1; the outer rate is r_E. Over the inner
interval of length 2 log m / (pi b / 3), the propagation
multiplier is 1. Over the exterior, it is r_E. Total: the
propagation of the physical partition over n blocks is
exp(-2 log m / (pi b / 3) * (pi b / 16)) = m^{-3/8}, plus
other factors. Section 4.8.
Step 8. The exterior operator may have Jordan blocks, so its
n-th power traces have polynomial-in-n corrections. Section
4.9 states the trace coefficient extraction lemma
(proved in Part 4.10) that separates off every peripheral
Jordan block and still recovers the physical value as a
single scalar. This is where "m^{1/6}" becomes a genuine
inequality.
Step 9. A homogeneous scalar normalization N(X) enters every
finite spin sum. At the homogeneous list X = (1,...,1), N(X)
has its own power of m, computed exactly from a Gram
determinant and a logarithmic statistic for even test
functions. Section 4.11.
Step 10. Combine the scalar normalization (Step 9), the
propagator estimate (Step 7), the trace lemma (Step 8), and
the exact value G_m(0) = 1 (which saturates the exponent).
Three-circle inequalities on a chain of disks across the
complex fugacity plane complete the proof. Section 4.12.
Localization (Section 5.4) then gives (1d) by comparing
G_m(2) to two independent planar partitions.
3.2 Theorem 2 -- finite chords and visit probabilities.
Step 1. The chord mass. Lemma 5.1 (the "convex-cap"
restriction) compares the chord sum inside D_R to the
half-plane arch mass. Proposition 5.1 (the arch estimate)
proves W_R = R^{-1/4+o(1)} for the total mass of half-plane
arches from a fixed port reaching radius R. The proof uses
an added-row expansion, so the boundary balance and a
positive mass moment determine each other.
Step 2. The wedge comparison (Section 5.2) gives the
matching upper bound: an arch reaching distance R leaves a
wedge of angle pi/3, and the lost mass in that wedge is
comparable to the arch weight. So W_R is bounded above
and below, and Z_R^chord = R^{3/4+o(1)} follows.
Step 3. The visit identity (Proposition 5.3). Remove the
triangle at a honeycomb vertex v. The signed balance in the
punctured domain is another exact cancellation. Comparing
the punctured and original balances shows: the raw visit
mass S_D(v) equals 3 Z_v + (corner weights) sum of
(z x^{|lambda|} times Z_out). Rearranging further relates
the ordinary visit mass sum_{gamma contains v} x^{N(gamma)}
to the sum of the two-face partitions Z_D(f) for f incident
at v. So visitors are comparable to nestings.
Step 4. Localization (Proposition 5.4). Use (1b) to drop
every cycle that travels far from its enclosed mark. The
remaining cycles live in a ball of radius C m around the
mark, and they project to two independent planar nestings.
Reverse inclusion gives the other direction. Conclude
Z(R) = R^{1/12+o(1)}.
Step 5. Combine: mean = (R^2 * Z(R/2)) / Z_R^chord
= R^2 * R^{1/12} / R^{3/4} = R^{4/3+o(1)}. Visit
probability = Z(R) / Z_R^chord = R^{-2/3+o(1)}. This
is (2b), (2c).
3.3 Theorem 3 -- unequal marked intervals.
Step 1. Physical identity (Proposition 6.1). Define
H^flip(X) as an explicit matrix element of the finite spin
data: two flips at sites i and j, an insertion, and a phase.
Then at the physical site list H^flip = H^poly. Proof in
Section 6.1: uses period-excluding walks on a finite
cylinder, and a careful phase computation for two marked
arcs.
Step 2. Generic flip formula (Proposition 6.2). For generic
nonzero sites, N(X)^2 H^flip has a clean closed form: a
single combinatorial sum over unmarked spins, with pair
factors F_s.
Step 3. Marked gas (Proposition 6.3). Run the same contour
+ bosonization machine, but with the site source changed to
account for the flips. The result is the same horizontal
gas with a marked, "kinked" angle.
Step 4. Suppression (Proposition 6.4). Near the two
physical site heights -f and 1-f, the site factors of
suppressed types are strictly below 1 with an exponential
margin. This gives exp(-c n e^{-kappa |x|}) suppression on
a scale |x| of order log n.
Step 5. Scalar cancellation (Lemma 6.2). The whole
site-dependent scalar normalizer N(X)^2 / S(X)^2 collapses
to a simple power m^{5/48-eps}, matching the homogeneous
normalization. This leaves H^flip bounded by a power of M
times the value of the gas limit at the marked data.
Section 6.5.
Step 6. Mixed spectral radius (Proposition 6.5). When one
list is longer, only that species's types are suppressed
on the far interval. The intermediate propagator allows
all types in species 1 and only unsuppressed types in
species 2. Its spectral radius is bounded separately
(Proposition 6.5). Section 6.6.
Step 7. Assemble (Section 6.7). Combine the outer rates
(Theorem 3's H^poly bound), the central power
(r_X^+ for the mixed region), the two peripheral gaps, and
the polynomial word-length factors. Every exponent cancels
to the m^{-4/3} * (n/m)^{1/3} of Theorem 3.
Step 8. Length-squared from Theorem 3 (Section 5.7).
Sum over ordered bond pairs. Choose "good" pairs (well-
spaced, with all three bond orientations nearby) so that
the marked cylinder of Theorem 3 embeds the polygon
injectively. Sum by dyadic distance d: d^2 pairs at radius
d, each bounded by m^{-4/3} (n/m)^{1/3} with m ~ d, n ~ H,
which collapses to H^{2/3+eps}. Bad polygons (too few good
bond pairs) are separately summable with exponential
decay. That proves (2d).
3.4 What to remember about the whole structure.
The signed transfer representation (Step 2-3 of Theorem 1)
is where "signed weights" appear. The crucial fact is that
the absolute values are summable (Step 5 of Theorem 1,
"stability"). That allows every Step after to be an honest
inequality on the positive physical partition, even though
the "answer" is computed by signed gas contour integrals.
Two independent computations exist inside the paper:
(a) the cylinder exponent m^{1/6} of Theorem 1, which
uses the full compact-propagator-plus-trace-extraction
machinery; (b) the two-bond estimate of Theorem 3, which
uses a somewhat different version of the same machinery
because of the marked bonds. Theorem 2(b), 2(c) follow
from (a); Theorem 2(d) from (b).
The paper has one "cheap" ingredient (the boundary
balance, which is the exact Duminil-Copin-Smirnov
cancellation applied to half-planes, hexagons, caps,
and punctured domains) and one "expensive" ingredient
(the signed transfer machinery). Almost every quantitative
result is a meeting point of the two.
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PART 4. PROVING THEOREM 1 -- THE CYLINDER EXPONENT
================================================================
PART 3 gave the map. This part walks through it in order.
Each subsection has one goal, one plain-words idea, and one
result for the next. Algebra lives in the original paper;
here we recite only the shapes of the formulas that matter.
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4.1 The oriented transfer and the height limit.
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Goal: turn the positive partition G_m(g) into a matrix element.
Seed: the boundary balance. In any convex region D, sum
first-exit walks from a fixed starting port, with weight
x^N for length N and phase e^{isT} for total turn T, where
s = 3/8. The signed sum over all walks ending on the
boundary equals 1 identically. This is the
Duminil-Copin-Smirnov cancellation; everything below
inherits its authority.
To build a cylinder from this seed, put three states on each
occupied honeycomb edge: empty, forward, backward. Attach a
phase to each turn, to each seam crossing (by twist p = i
above the cut and p = -i below), and to each crossing of
the "A cut". The result is a local transfer matrix R(phi)
with five trigonometric entries (u, d, v, a, b). At
phi = alpha = pi/8 those five weights are exactly the
honeycomb weights (x, x^2, x^2, x^2, 0).
Now orient every loop of a closed finite cylinder either
counterclockwise (contractible) or eastward (wrapped, so
both marks lie on its left). The turn phase is -i; the
seam phase splits as p^{-I_1} on one arc and 1 - I_1 on
the other; the A-cut phase is (ih)^{I_2 - I_1}, where I_1
and I_2 say whether O and O' lie on the interior (or
upper) side. Multiplying and summing over the two
orientations gives
(-i)(-1)^{I_1} (ih)^{I_2 - I_1}
plus its inverse. The sum is 0 unless I_1 differs from I_2
(that is, the loop separates O from O'), and in that case
the two terms combine to h + h^{-1} = g. Therefore the
signed sum over loop families equals the positive partition
G_m(g). Signed transfer, positive result.
The vacuum limit. Grow the cylinder height in both
directions away from the cut. Arches reaching from a rim
to the middle die off (the arch summability proved later
in 5.1), so the top and bottom rims decouple. The height
limit gives two vacuum vectors
L_-^phys = bottom vacuum,
R_+^phys = top vacuum,
indexed by spin configurations on the cut. The partition
becomes
G_m(g) = L_-^phys * D_A * R_+^phys,
where D_A = (ih)^{sum over sites of A of s_j} is the
insertion along the cut.
Result: G_m(g) is a matrix element of two vacuum vectors
against the insertion D_A.
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4.2 Exchange, fusion, and the local projector.
----------------------------------------------------------------
Goal: two algebraic identities. One lets rows be reordered.
The other lets adjacent rows collapse.
Exchange. The transfer matrix satisfies the Yang-Baxter
equation
R_12(t) R_13(t+phi) R_23(phi)
= R_23(phi) R_13(t+phi) R_12(t),
as a meromorphic identity in the two parameters. In plain
words: rows can be permuted freely without changing any
matrix element. Consequence: a family of rows with a common
twist has a simultaneous eigenbasis.
Fusion. For delta = 2 or 3, there is an injective embedding
J_delta from a smaller space E_delta into V tensor V:
delta = 2. E_2 three-dimensional, spins -1, 0, +1.
J_2 |0> = |00> + x sum_k lambda^k |k,-k>,
J_2 |k> = x (|k,0> + |0,k>), k = +/-1.
delta = 3. E_3 one-dimensional.
J_3 (1) = sum_k lambda^k |k,-k>,
with lambda = e^{-2 i alpha}. The embedding satisfies
R_23(phi) R_13(phi + delta*alpha) (J_delta tensor I)
= (J_delta tensor I) * R_f(phi),
with R_f(phi) = R(phi + alpha) for delta = 2, and R_f = I
for delta = 3. In plain words: two rows at rapidities phi
and phi + delta*alpha collapse to one row (delta = 2) or to
nothing (delta = 3) on the embedded subspace. On the
quotient there is a simple zero at the fusion point.
Both identities are local. Lift each over the whole cut and
the vacuum pairing factors through the image of J_delta per
site. The 3^M spins in V^{tensor M} thus reduce to (3 or
fewer)^M spins per block.
Why this matters: exchange makes rows freely movable;
fusion makes them fully collapsible. Between the two,
vacuum vectors can only see a small per-site label.
----------------------------------------------------------------
4.3 The finite spin sum.
----------------------------------------------------------------
Goal: reduce the vacuum pairing to a finite sum over a
single spin in {-1, 0, +1} per site.
Insert at each site a spectral projector Pi_p for the twist
p = i on the group above the cut and p = -i on the group
below. Because the fusion algebra is so rigid, the projector
has a closed determinant form,
Pi_p = det L(T) / det L(1),
where L(T) is a matrix in the commuting row coefficients T_j
obtained by expanding the fusion equations, and det L(T)
vanishes identically on the common fixed space.
The denominator is a Pfaffian:
det L(1) = (Pf S(Y))^2,
where S is an explicit skew kernel and Y is the alternating
list Y = (x_1, q x_1, ..., x_m, q x_m), q = e^{2 i alpha}.
The square N(X)^2 clears the poles of the two selected
vacuum vectors L_- and R_+.
Inserting the projectors and applying fusion gives the
finite spin sum formula: it expresses N(X)^2 G_{A,B}(h+h^{-1})
as
sum over (s_1, ..., s_M) in {-1, 0, +1}^M
with sum_{j in A} s_j = sum_{j in B} s_j = Q
of h^{Q} times a product of site and pair factors.
Structural facts about the sum:
- It is symmetric within each group A, B.
- Individual degree in any one site variable is at most
2(M-1).
- Fusion of two sites at ratio q^2 or q^3 reduces it to a
smaller sum of the same form.
- At zero spin per site, all factors collapse to 1.
Crucial nonvanishing. Let S(X) be the explicit Pfaffian
prefactor. At the physical list X = (1, ..., 1), the ratio
N(X) / S(X) is bounded below by (2/3)^{M/2} M^{5/48 - eps}
for every eps > 0 and all sufficiently large M. Hence
N(X) does not vanish anywhere near the physical list, so
dividing by N(X)^2 is safe.
The partition has become a finite sum of explicit
polynomials.
----------------------------------------------------------------
4.4 Contours and the angular trace.
----------------------------------------------------------------
Goal: replace the finite spin sum by a contour integral, and
then by a fermionic trace.
Problem. Individual summands of the finite spin sum have
poles when two site variables coincide. The combined sum is
analytic (it equals the partition), but the terms are not.
Fix. For each species (group A and group B) and a small
radius r > 0, wrap each site's variable onto a circle
gamma_r of radius r about the origin, turning each residue
into a contour integral. Two residues hitting the same site
cancel (their first Pfaffian fields coincide), so the
expansion terminates after m currents per species.
Bilinear identity. The contour pattern is exactly a
fermionic Wick trace on an angular space of circumference
T (a length parameter, taken to infinity at the end).
Introduce anticommuting fields psi(v) with two-point
function C^{[T]}(v - u), periodic-antiperiodic on the
angular circle. The regulated contour expression reads
C_{sigma,T}(W)
= < exp( sum_delta int_gamma_r dw/(2 pi i)
(-z^delta) e^{i delta phi_sigma(w + i delta/2)}
prod_l U_delta,T(w - w_l)
psi(w + 3i) psi(w + i delta) ) >_T .
The fields phi_sigma come from an external Gaussian regulator;
U_delta,T is the periodized product of tilts; z = e^{i theta}
is the physical twist; delta = +/-1 labels the deflection.
Simplify. Take the regulator average away and define
C_{m,T} as the full normalized contour value at W = 0^m
(all m sites coalesced). Proposition "fixed-site regulated
Pfaffian":
C_{m,T} -> (N(X)^2 / P_0) * G_m(h + h^{-1}), T -> infinity,
locally uniformly in theta, at each fixed site count m,
where P_0 is the all-zero-spin term Pi_0(W) of the finite
sum. In one line: the contour current recovers the partition
in the infinite-period limit at fixed m.
Next step: bosonize the trace.
----------------------------------------------------------------
4.5 Bosonization: the horizontal gas.
----------------------------------------------------------------
Goal: rewrite the angular trace as a signed gas of particles
on a horizontal circle of length T.
Charge sectors. The fermion system has three zero-mode
charges c_1, c_2, c_3. Nonzero oscillator modes fill each
charge sector. The seam operator W_b from the contour
expression (an ordered product of exponentials of currents)
shifts one zero-mode charge by an integer ell and swaps two
sector labels. Trace closure forces
S_1 = ell, n_2 = (F_3 - E_2 - ell)/2,
n_3 = (-F_3 - E_2 - ell)/2,
where (S_1, E_2, F_3) are the total inserted charges of the
three sheet types and n = (n_1, n_2, n_3) is the ground
charge just to the right of the inserted fields. Both
displayed values are integers by charge parity, so the trace
is well-defined.
Particle decomposition. Split the fermions into "short" and
"outer" field types, then evaluate the H-dress that comes
from the seam contraction. The result is six basic particle
types plus one fused type. Each has a short height, an outer
height, an H-charge, a charge jump, a local activity gamma,
and a site multiplier omega_v(x) acting at the m-th power:
type charge omega_v(x) at |x| large
P +1 B(x) < 1
M -1 1 1
O_+ 0 B(x - i) < 1
O_- 0 B(x + i) < 1
U -1 B(x + i(3/2+d))^{-1} < 1
L -1 B(x - i(3/2+d))^{-1} < 1
F_pair -2 1 1
with B(v) = (cosh(pi v/3) - sqrt(3)/2) /
(cosh(pi v/3) + sqrt(3)/2).
The meromorphic identities B(v+i) B(v-i) = B(v) and
B(v+3i) = 1/B(v) keep the seven-type list closed under
collisions. Only M and F_pair have site multiplier exactly
1 at every x; all six basic activities are explicit
(sqrt(2)/4 times exponentials of a fixed kernel).
Fusion of U and L. Two same-species U and L can collide;
the residue at collision is the fused particle F_pair with
local activity
gamma_{F_pair} = (9 pi^2 / 8) exp( K_diag(0) + K_diag^reg(3i) ),
a positive constant independent of T, m, theta. Counting
collisions uses the binomial fraction
1/((u-r)! (l-r)! r!), where u, l are current counts of U
and L and r is the number of fusions.
Exact gas identity. Let Z_{m,T} be the sum over
configurations Y (distinct real positions on R/T Z, labeled
by the seven types) times a state weight W_{d,theta}(Y, p).
Here p is a running charge vector obeying
p(T) = p(0) mod 8 Z(1, -1), climbing by -q_v at each
particle; the pair interaction R_{uv}(x) decays
exponentially with x. Then
C_{m,T} = c(T) exp(m^2 kappa_T) Z_{m,T},
where c(T) is an explicit scalar (empty oscillator trace,
modular prefactor, Poisson sum over winding), and kappa_T is
an explicit scalar rate with
lim_{T to infinity} kappa_T = 2 log(2(2 + sqrt(2))/9).
Both c(T) and kappa_T have closed-form period limits and are
subexponential in T.
Sign of the story so far. The positive partition is now a
signed gas Z_{m,T}. Because Z is signed, we still cannot
bound it directly. The next subsection (4.6) proves that
the absolute value of every gas weight decays quadratically,
so the signed sum converges absolutely.
----------------------------------------------------------------
4.6 Absolute stability.
----------------------------------------------------------------
Goal: bound the modulus of every gas weight, so the signed
sum Z_{m,T} is absolutely convergent.
Statement (Theorem "absolute stability"). At a fixed
displacement d with 0 < |d| < d_0 (small), fix a compact set
Theta of angles theta. There are constants c > 0 and
C < infinity, depending on d and Theta, such that for every
finite configuration Y of particles on R/T Z and every
closure-compatible state path p,
|W_{d,theta}(Y, p)|
<= exp( C T - c sum_{j=0}^{T-1}
( n_j^2 + p_{c,j}^2 ) ),
where n_j = number of particles in box [j, j+1) and
p_{c,j} = (p_1(j+) + p_2(j+)) / sqrt(2) is the running
common charge just to the right of that box.
Reading. Every extra particle in a box costs at least
quadratically. Every extra unit of running charge at a box
boundary costs at least quadratically. The cost dominates
at the exponential level: the more concentrated the gas,
the more it is punished.
How it is proved. The gas weight splits into five height
kernels -- short, outer, H, fusion residue, and the state
term -- each with explicit form. The combined height matrix
H(k) has a coercivity structure:
- near k = 0, the leading part is
( a_c / k ) * c(k) c(k)^T
for the vector of particle charges c_v, plus O(k) times
a strictly positive matrix on the quotient by the
linear relations among P, M, O_+, O_-;
- at large k, H(k) converges exponentially to a positive
matrix, so there is a uniform spectral gap on the
quotient by that null subspace;
- throughout, H(k) is positive on count-respecting
directions, with a quantitative gap of size min(k, 1)
times a fixed positive constant.
Convert to real space via Fourier inversion. Each k-mode
becomes a real-space kernel (2 pi / |k|) times a symmetric
function of |k|. The leading charge term becomes a Poisson
kernel (positive, long-range). Inserting a Gaussian
smoothing of width sqrt(T) in space and applying a discrete
Young inequality turns the sum of real-space kernels into
the box-count and running-charge squares displayed above.
Consequence. Z_{m,T} converges absolutely, uniformly in
theta in Theta at each fixed period T. The sum over
particle count converges like a Gaussian:
sum_n A^n exp(-c n^2) < infinity for any finite A and
positive c. The signed gas is therefore a genuine function
of T and theta with controlled analyticity.
This justifies every later coefficient extraction from
power traces of the compact propagator (4.7).
----------------------------------------------------------------
4.7 A compact propagator.
----------------------------------------------------------------
Goal: turn the gas sum into a compact trace, so the
trace-extraction lemma (4.8) applies.
Block partition. Let b be a large fixed block size and put
T = b n for an integer n. Group R/T Z into blocks of length
b, numbered j = 0, ..., n-1. Each block has a state
S_j = (Y_j, p_j): a particle configuration plus its running
charge.
Successor rule. The next block starts from p_j - q(Y_j),
where q(Y_j) is the block's charge. The gas weight
factorizes as a product of a local kernel w_theta(S_j, S_j+1)
over blocks, plus a "tail" factor for pairs at block
distance at least 2. The tail factor is exactly the far-side
interaction. It has an analytic Bargmann representation
using a nilpotent Jordan matrix A_0 whose diagonal entries
are pi l / 6 (l >= 1) and whose nilpotent parts are
uniformly bounded.
Gauge. The local kernel has a gauge freedom: replace
w_theta(S, S') by exp(Phi(S') - Phi(S)) * w_theta(S, S')
for any real function Phi; products around closed cycles
are unchanged. Choose Phi so that the gauged kernel
satisfies
|w_theta(S, S')|
<= exp( C_b - delta_b ( |Y_j|^2 + |Y_{j+1}|^2 + p_c^2 ) ),
uniformly in theta in a compact set Theta. This is what
makes everything that follows uniform.
Properties. The gauged kernels are Hilbert-Schmidt
uniformly in theta in Theta, and holomorphic in
Hilbert-Schmidt norm. The trace of a product of n blocks
equals the gas sum Z_{m,bn}. The Bargmann representation
handles the far-side tails exactly, so no approximation
enters.
Empty-block traces. Let E_0 be the "all seven types
allowed" operator at theta and I_0 the "only M and F_pair
allowed" operator. Both are parametrized by theta. Then
tr(E_0^n) = det(I - A_0^n)^{-1} / c(bn),
tr(I_0^n) = 8 det(I - A_0^n)^{-1} sum_{k in Z} u_k^n,
with
u_k = exp( -4 pi b / 9 * (k + theta/pi)^2 ).
The factor 8 is the mod-8 winding multiplicity: each
k = p_1 + p_2 has eight representatives in
Z^2 / 8 Z(1, -1).
Spectral radii. Reading off these traces gives
r(E_0) = r_E = e^{-b pi / 16} (all seven types),
and the leading eigenvalue of I_0 is u_0 = 1. For
|Re theta| < pi/2, u_0 is semisimple: the Jordan blocks of
the zero-charge region do not affect it.
How this feeds 4.8. The site factor |B(x)|^m < 1 at every
x off the origin, so the m-th power of the site factor
suppresses all six basic particles except M and F_pair,
with exponential rate pi/3 in |x|. Inside an interval
|x| < L ~ log(m) / (pi b / 3), only M and F_pair survive at
all: the operator I_0 acts. Outside, all seven types are
present: the operator E_0 acts. The two operators are
exactly the ones we just computed.
----------------------------------------------------------------
4.8 Trace extraction with Jordan blocks.
----------------------------------------------------------------
Goal: extract the physical value R from a trace
difference, even when the exterior operator has peripheral
Jordan blocks.
Setup. Let E be a Hilbert-Schmidt operator of spectral
radius 1. Its peripheral eigenvalues are ζ with |ζ| = 1.
Write P_ζ for the Riesz projection onto ζ and J_ζ for the
nilpotent part, so on ran(P_ζ),
E = ζ P_ζ + J_ζ, with J_ζ^{q_ζ} = 0
for some finite q_ζ.
Problem. The bare trace tr(E^n P_ζ) = ζ^n d_ζ + O(ρ^n)
only sees the algebraic multiplicity d_ζ. The nilpotent
part J_ζ is invisible in this limit.
Setup for lack of information. Introduce two word families
W+, W- with integer lengths and exponentially decaying
tails:
sum_{W±} e^{ε l(W)} ||W|| < infinity.
Define the shifted trace
F_s = sum_{l(W+) + l(W-) <= s}
tr( C W+ E^{s - l(W+) - l(W-)} W- ).
Assume F_s - R tr(E^{s+a}) -> 0 for some scalar R and
integer a >= 0. We must recover R from F_s.
The lemma. Define a polynomial in a formal variable z by
P_ζ(z) = sum_{W+, W-} ζ^{-l}
tr( C W+ [ sum_{t=0}^{q_ζ - 1} binom(z - l, t) ζ^{-t}
J_ζ^t P_ζ ] W- ),
where l = l(W+) + l(W-) and binom(z - l, t) is the
binomial polynomial. Then
P_ζ(z) identically equals R * d_ζ * ζ^a.
Hence
R = P_ζ(0) / (d_ζ ζ^a),
|R| <= K * ||C||_1,
where K depends only on E, ε, the two weighted word sums,
and the nilpotent orders q_ζ. It does not depend on C or a.
In plain words: the polynomial identity is exact. Reading off
the coefficient of any peripheral eigenvalue gives R
directly. Every Jordan block is accounted for by the
binomial polynomials binom(z - l, t).
Special case a=0, no words. If W+ = W- = identity and
F_s = tr(C E^s), the identity says tr(C P_ζ) = R d_ζ ζ^a.
This is the elementary case; the lemma extends it.
How it applies. In the cylinder problem:
E = E_0 / r_E (the normalized "all types" operator);
C = C'_m = (u / r_E)^{2L} C~_m (central block product);
a = 2L;
d_ζ = algebraic multiplicity of ζ on E'.
Then R is the limit of the normalized partition trace, and
|R_m| <= C |u / r_E|^{2L},
uniformly in m, uniformly in theta in a fixed compact set.
Profile form. The central product C~_m has an analytic edge
profile. In the notation a_m = m e^{-k_1 b L} in
[1, e^{k_1 b}],
R_m (u / r_E)^{-2L} d_ζ ζ^{2L}
= F_ζ(a_m, theta) + O(e^{-cL}),
where F_ζ is analytic in a near [1, e^{k_1 b}] and locally
analytic in theta. This is what the assembly (4.10) plugs
in.
----------------------------------------------------------------
4.9 The Pfaffian normalizer N(1, ..., 1).
----------------------------------------------------------------
Goal: compute N(X) at the homogeneous physical list
X = (1, ..., 1) of size 2m.
Recall the definition. With
P(X) = prod_{j<k} d(x_j, x_k) / (x_j - x_k),
d(u,v) = u^2 + sqrt(2) u v + v^2,
define
N(X) = P(X) * Pf_{j,k} [ (x_j^2 - x_k^2) / d(x_j, x_k) ].
The coincident-site value is by polynomial continuation.
Result.
N(1, ..., 1) = (2 p_0 / 3)^{m(2m-1)}
* (2/3)^m
* m^{5/48 + o(1)},
with p_0 = 2 + sqrt(2).
Idea. Compare with a reference Pfaffian at a different
kernel,
C(w) = tanh(pi w / 6),
whose Pfaffian is exactly computable by Schur's Pfaffian
identity:
Pf_{j,k} [ (y_j - y_k) / (y_j + y_k) ]
= prod_{j<k} (y_j - y_k) / (y_j + y_k).
Taking y_j = exp(pi w_j / 3) recovers Pf[C(w_j - w_k)] as a
full product.
As the w_j tend to 0,
[d(x_j,x_k) / (x_j - x_k)] * C(w_j - w_k) -> 2 p_0 / 3,
so P(X) * Pf[C(w_j - w_k)] gives the base factor
(2 p_0 / 3)^{m(2m-1)}.
The remaining ratio of two confluent Pfaffians becomes a
ratio of Gram determinants of the functions
1, k^2, k^4, ..., k^{2m-2} against two weights:
B0_hat'(k) = 8 k cosh(k) / sinh(4k),
C_hat'(k) = 6 k / sinh(3k).
Their ratio is
B0_hat'(k) / C_hat'(k) = (2/3) e^{phi(k)},
with phi(k) = log[ (1+t^2)(1-t^6) / (1-t^8) ], t = e^{-k}.
The factor (2/3)^m comes out; the remaining ratio is a
Gram determinant of e^{f(u)}, with u = 3k/pi and
f(u) = phi(pi u / 3).
Log statistic. Let mu(u) = 2 u / sinh(pi u) (a probability
density, with Fourier transform 1/cosh^2(s/2)). For f even,
continuous and exponentially decaying,
log det( P_m e^f P_m | on ran P_m )
= (log m) / (2 pi) * int_R f(u) du + o(log m),
where P_m is the L^2(mu) projection onto the even
polynomials of degree at most 2m-2. The proof diagonalizes
the projection kernel by Meixner-Pollaczek polynomials and
uses the Gaussian tail bound to control errors.
In our case, expanding the logarithms as geometric series,
int_0^infty phi(k) dk
= pi^2/24 - pi^2/36 + pi^2/48
= 5 pi^2 / 144,
so
int_R f(u) du / (2 pi)
= (3 / pi^2) * (5 pi^2 / 144)
= 5/48.
Therefore the Gram ratio is m^{5/48 + o(1)}, giving
N(1, ..., 1) as claimed.
Why the exponent 5/48 matters. It enters the partition
normalization as N(X)^2, which is 2 * (5/48) = 5/24.
That is the exact factor the assembly (4.10) subtracts
from the central+exterior rate 3/8, giving 3/8 - 5/24 = 1/6.
----------------------------------------------------------------
4.10 Assembly: G_m(2) = m^{1/6+o(1)}.
----------------------------------------------------------------
Goal: put together 4.3 (spin sum), 4.7 (propagation
rates), 4.8 (trace extraction), and 4.9 (Pfaffian) to prove
the cylinder exponent.
Angular parametrization. Set g = 2 cos Θ. The polynomial
G_{A,B}(g) has positive coefficients and satisfies
G_{A,B}(0) = 1 (empty family). Its complex continuation
enters the bosonization through h = e^{-iθ} with Θ = -2θ.
The exponent identity. Three pieces combine:
(a) central + exterior propagation rate (4.7):
3/8 - 8 θ^2 / (3 pi^2)
in the θ variable. With Θ = -2θ, this is
3/8 - 2 Θ^2 / (3 pi^2)
in the Θ variable.
(b) scalar normalizer (4.9), entering as N(X)^2 (not N):
5/24.
(c) trace extraction (4.8) turns the difference into a
bound on |R_m| ~ m^{p_*(Θ)}, with
p_*(Θ) = (a) - (b) = 1/6 - 2 Θ^2 / (3 pi^2).
Statement. Uniformly for complex Θ in a compact subset of
|Re Θ| < π,
|G_{A,B}(2 cos Θ)| <= m^{Re p_*(Θ) + o(1)}.
Proof in one line. Sections 4.3--4.9 establish the bound.
4.9 gives the closed form for N(X)^2. 4.7 gives the rate.
4.8 makes the extraction valid even when E_0 has
peripheral Jordan blocks.
At the two endpoints.
Θ = 0 (g = 2): p_*(0) = 1/6, so |G_m(2)| <= m^{1/6+o(1)}.
Θ = pi/2 (g = 0): p_*(pi/2) = 1/6 - 2 (pi^2/4)/(3 pi^2) = 0,
and G_{A,B}(0) = 1 exactly. So the bound is saturated.
Lower bound at g = 2. Suppose G_{A,B}(2) <= m^{1/6 - δ} for
some fixed δ > 0. By positivity of the coefficients,
|G_{A,B}(g)| <= G_{A,B}(2) for |g| <= 2, so on a small
positive-Θ disk we have
G_{A,B}(2 cos Θ) m^{-p_*(Θ)} <= m^{-δ}.
The three-circle inequality along finitely many overlapping
disks from that disk to Θ = pi/2 propagates a positive
fraction of the gap to the endpoint. But there
G_{A,B}(0) = 1 is exact; contradiction. Hence the lower
bound matches, and G_m(2) = m^{1/6+o(1)}.
Logarithmic derivative at g = 0. At Θ = pi/2 the exponent
p_*(Θ) vanishes. Since the analytic factor F_ζ(a, θ) is
nonzero on the closed a-interval [1, e^{k_1 b}] (by
compactness and a small open neighborhood of θ_i = -pi/4),
differentiating the analytic expression along the local
branch Θ(g) near g = 0 gives
-d^2/dg^2 log G_m(0) = O(log m).
Together with G_m(2) = m^{1/6+o(1)} this proves Theorem 1
parts (1a) and (1b).
----------------------------------------------------------------
4.11 Localization: Z(R) = R^{1/12+o(1)}.
----------------------------------------------------------------
Goal: prove the planar nesting exponent from the cylinder
exponent.
Setup. G_m(2) counts families of disjoint cylinder cycles
separating the two marks O and O'. Each cycle is either
contractible (its disk lift encloses exactly one mark) or
noncontractible (wraps and separates the two ends).
Reduction to planar. The regular hexagon D_R sees the same
partition locally: same disjointness, same weights. The
bridge is that G_m(2) counts configurations where some
cycles stray far from their mark. We have to show that
family is negligible.
Bad cycles (Lemma "escape"). Fix a constant C > 1. Drop:
- all noncontractible separating cycles;
- all contractible cycles whose disk reaches
|x_horiz| >= M/4 or has diameter > CM.
Each dropped family contributes O(sqrt(I_M)), where I_M is
the incompatible-pair curvature sum, which 4.10 identifies
as O(log M). Total dropped weight is O(sqrt(log M)) =
M^{o(1)}.
Remaining cycles. Every surviving cycle's disk lives in a
closed ball of radius CM about its enclosed mark. Same-mark
cycles are strictly nested (their disks have strict
diameter inflation). Opposite-mark cycles are disk-disjoint.
Hexagon comparison. A regular hexagon of side ⌈C'M⌉ for
C' > 2C/sqrt(3) contains this ball. Same-mark disjointness
is preserved on the chosen lifts. Dropping constraints
between the two marks then bounds the remaining partition
by the product of the two planar ones:
G_m(2) <= Z(⌈C'M⌉)^2 * exp(O(sqrt(log M))).
For the other direction, hexagons of side ⌊cM⌋ (with
0 < c < 1/4) about the two marked lifts embed into the
cylinder with all disjointness preserved. So
Z(⌊cM⌋)^2 <= G_m(2).
Combine. Using G_m(2) = M^{1/6+o(1)}:
Z(⌊cM⌋)^2 <= M^{1/6+o(1)},
so Z(R) <= R^{1/12+o(1)}.
Z(⌈C'M⌉)^2 >= M^{1/6 - o(1)},
so Z(R) >= R^{1/12 - o(1)}.
The change of variable R ~ cM (or R ~ C'M) preserves the
power up to o(1). Therefore Z(R) = R^{1/12+o(1)}.
Result: Theorem 1 part (1d). Together with 4.10, Theorem 1
is proved.
End of PART 4.
================================================================
PART 5. PROVING THEOREM 2 -- CHORDS, VISITS, LENGTH^2
================================================================
PART 5 takes Theorem 1 and delivers Theorem 2. The engine is
the arch estimate (5.1), which controls the mass of long
walks on the boundary of a region. Chords (5.2), visit
identities (5.3), mean lengths (5.4), and the length-squared
bound (5.5) all follow.
----------------------------------------------------------------
5.1 The arch estimate W_R = R^{-1/4+o(1)}.
----------------------------------------------------------------
Goal: bound the mass of half-plane first-exit arches.
Definition. In the half-plane bounded by a horizontal
tessellation line, let W_R be the total critical weight of
first-exit arches from a fixed port that reach Euclidean
distance at least R from that port.
Result: W_R = R^{-1/4+o(1)}.
The proof has four sub-steps.
(a) Boundary balance. In any convex region D, sum first-exit
walks from a fixed starting port with the same phases as in
4.1. That signed sum is 1. Applied to the half-plane, this
gives the half-plane saturation
2 sin(pi/8) * sum_{d>0} H(d) = 1,
where H(d) is the total half-plane arch weight from a
fixed port to the port d boundary steps to its right. The
line d=0 is excluded (first-exit arches have positive
length). Positivity bounds the left side by 1, so the sum
converges.
(b) Added-row fusion. Introduce a finite cylinder of
circumference M, and put one new outer rhombus row on top
(L honeycomb rows). Subtract the partition without that
row. The increment F(v) is a polynomial in the row weight v,
of degree at most M. Local fusion (4.2) gives a functional
relation F(v) F(q^2 v) = F(q v) at each site, and following
the linear coefficient and the top coefficient through that
relation pins down
F'_v(1) = M^{3/4+o(1)}.
The exponent 3/4 comes from a Pfaffian determinant law on
ordered points k_1 < ... < k_M with weight function
w(k) = 2 (1 + 4 sinh^2(alpha k)) / sinh(pi k),
combined with a Gamma comparison (or equivalently a von
Mises ensemble). Independent of the boundary balance, this
computation is the deepest piece of PART 5.
(c) Deficit. The increment's coefficient of v^{M-1} is
1 - c_alpha * sum_{d=1}^{M-1} H_M(d), where
c_alpha = 2 sin(alpha). Hence the deficit
delta_M := (1/c_alpha) - sum_{d=1}^{M-1} H_M(d)
= F'_v(1) / (c_alpha M)
= M^{-1/4+o(1)}.
Here H_M(d) is the cylinder version of the arch (arches
disjoint from all their M-translates). Taking M to infinity
shows sum_{d>0} H(d) = 1/c_alpha exactly, and the tail
A(M) := sum_{d >= M} H(d)
<= delta_M = M^{-1/4+o(1)}.
(d) Radius vs. length. Every arch contained in a half-disk
of radius M/2 around its source is M-translate-disjoint, so
such arches are fully counted in H_M(d). Therefore the
missing mass delta_M is exactly the mass of arches reaching
distance at least M/2, and
delta_M <= W_{M/2}.
Choosing M slightly above 2R gives the lower bound
W_R >= R^{-1/4+o(1)}.
The upper bound uses a wedge comparison. Intersect the
half-plane with a wedge of angle pi/3 (bounded by the source
ray and its reflection). Signed balance on the wedge gives
a linear relation between the lost left arch weight
L_<(a) and the lost right arch weight L_>(a). Summing over
a finite range and using the balance identity gives
L_<(a) + L_>(a) <= a^{-1/4+o(1)}.
Two reflected wedges whose vertices are at horizontal
distance a on opposite sides of the source intersect in a
diameter-O(a) region. Any arch reaching distance Ca for a
sufficiently large fixed C must leave at least one of the two
wedges, so its weight is at most the sum of the two wedge
losses. Hence W_R <= R^{-1/4+o(1)}.
Together: W_R = R^{-1/4+o(1)}.
----------------------------------------------------------------
5.2 Chords from arches.
----------------------------------------------------------------
Goal: bound Z_R^chord = R^{3/4+o(1)}.
Setup. D_R is the regular hexagon of side R. C_R is the
set of ordered first-exit chords of D_R (a self-avoiding walk
from one boundary port to another, staying inside D_R, whose
last vertex is a boundary port other than the start) with
diameter at least R/10. Z_R^chord = sum over gamma in C_R
of x^{N(gamma)}. The sum runs over every start and every
exit, retaining the direction from start to exit.
Upper bound (fixed source). Fix one source port a. A chord
of diameter R/10 must reach at least R/20 from a. Choose a
small hexagon cap P of diameter about R/25 inside D_R that
contains a but is well clear of any such distant exit. By
the convex-cap lemma (below), the chord weight lost by
restricting D_R to D_R ∩ P is bounded by a constant times
the weight of artificial exits (first-exit walks ending on
the new boundary of D_R ∩ P). Those artificial exits are
half-plane arches from sources within P, whose total is at
most C W_{cR}. Therefore the total chord weight from a
fixed source is at most C W_{cR} = R^{-1/4+o(1)}.
Lower bound (fixed source). Choose sources in a fixed
middle portion of one side of D_R, at distance at least
cR > R/10 from every other side. A half-plane arch from
such a source reaching radius R is either a macroscopic
chord contained in D_R, or else is lost by the convex-cap
argument. The lost weight is again at most C W_R. So the
macroscopic chord weight from each such source is at least
a constant times W_R = R^{-1/4+o(1)}.
Combine. There are O(R) sources in the middle portion.
Summing the fixed-source bounds:
c * R * R^{-1/4+o(1)} <= Z_R^chord
<= C * R * R^{-1/4+o(1)}.
Therefore
Z_R^chord = R^{3/4+o(1)}, which is Theorem 2 part (2a).
Convex-cap lemma in one line. Let K ⊂ D be finite convex
tessellation polygons sharing a starting port. The raw weight
of first-exit chords of D that are lost when restricting to K
is at least sin(alpha) times the artificial-exit weight E of
K and at most E / sin(alpha). Proof: both sides are signed
balances; the two sums coincide on common chords (which
cancel), and every real coefficient in either convex domain
sits between sin(alpha) and 1; taking real parts gives the
two-sided bound.
----------------------------------------------------------------
5.3 The visit identity.
----------------------------------------------------------------
Goal: compare the raw weight of chords visiting a fixed
interior vertex v with a nesting partition.
Setup. D is a finite convex tessellation polygon. v is a
honeycomb vertex not adjacent to any boundary port. Let
S_D(v) = sum over first-exit boundary chords gamma of D
that visit v
of x^{N(gamma)} cos(3 T(gamma)/8).
Let Z_v be the sum over collections of disjoint unoriented
cycles that strictly enclose v (and avoid v), each cycle
contributing 2 x^{|lambda|}. Let Z_out(lambda) be the
analogous sum over disjoint cycles that surround lambda and
are disjoint from it. At a vertex lambda ∩ v, call the
corner convex if the interior angle is 2 pi/3 and reflex if
4 pi/3, and put
c_lambda = sqrt(2) (v convex corner of lambda),
= 2 + sqrt(2) (v reflex corner).
Identity.
S_D(v) = 3 Z_v
+ sum over cycles lambda through v of
c_lambda * x^{|lambda|} * Z_out(lambda).
Reading. The first term counts cycles that avoid v (as a
background). The second term counts cycles through v (each
occupies a corner at v).
The three incident faces. Let Z_D(f) denote the nesting
partition around a face center f. Then
sum over the three incident faces f of v: Z_D(f)
= 3 Z_v + sum over lambda through v of
2 k_lambda * x^{|lambda|} * Z_out(lambda),
where k_lambda = 1 at a convex corner and 2 at a reflex
corner.
Comparing the two identities gives
(1 / sqrt(2)) * sum over incident f: Z_D(f)
<= S_D(v)
<= sum over incident f: Z_D(f).
And boundary positivity gives
S_D(v) <= sum over gamma visiting v: x^{N(gamma)}
<= S_D(v) / sin(alpha).
So the ordinary visit mass is comparable to the sum of the
three incident-face nestings.
Proof idea in one line. Remove the triangle centered at v,
creating three inner ports. The signed balance in the
punctured domain still holds. Subtracting the punctured
from the original gives the difference as paths from the
three inner ports to outer exits. Take real parts and
reverse paths to arrive at v. Grow a tree from each inner
port: it ends (a) at an outer exit (giving S_D(v)),
(b) at first self-collision not enclosing v (cancels by
reflection symmetry), (c) at first collision with a
background cycle (cancels with the alternative first-contact
class), or (d) at an inner-inner return that closes a
cycle lambda through v (contributes c_lambda times Z_out).
The coefficient c_lambda is the residual turn phase of that
closed lambda.
----------------------------------------------------------------
5.4 Mean chord length and bulk visit probability.
----------------------------------------------------------------
Goal: prove Theorem 2 parts (2b) and (2c).
Setup. D_R and C_R as in 5.2. Let Z(R) be the planar +2
nesting partition around the central face center in D_R.
For every honeycomb vertex v in D_R,
V_R(v) = sum_{gamma in C_R, v in gamma} x^{N(gamma)}.
The mean chord length is
E_R^chord N = (1 / Z_R^chord) sum_{v in D_R} V_R(v).
Bulk vertices. For v at distance at most R/10 from the
center, every C_R chord through v is macroscopic. A
translated hexagon of side floor(R/2) centered at each
incident face fits inside D_R. Positivity of the visit bound
from 5.3 gives
c Z(floor(R/2)) <= V_R(v)
<= C Z(3R),
uniformly in the stated range.
Summation. For any vertex v not adjacent to a boundary port,
a translated hexagon of side 3R centered at an incident face
contains D_R, giving the upper V_R(v) <= C Z(3R). There are
O(R^2) bulk vertices and only O(R) boundary-adjacent
vertices, each contributing at most one chord to the
normalized mean (a chord visits any vertex at most once).
Hence
c R^2 Z(floor(R/2)) / Z_R^chord
<= E_R^chord N
<= C R^2 Z(3R) / Z_R^chord + C R.
Plugging Z(R) = R^{1/12+o(1)} (4.11) and
Z_R^chord = R^{3/4+o(1)} (5.2):
R^2 * R^{1/12} / R^{3/4}
= R^{2 + 1/12 - 9/12}
= R^{2 - 8/12}
= R^{4/3}.
Therefore E_R^chord N = R^{4/3+o(1)}, which is (2b).
Visit probability. For any v at distance at most R/10 from
the center, divide V_R(v) by Z_R^chord:
V_R(v) / Z_R^chord
= R^{1/12+o(1)} / R^{3/4+o(1)}
= R^{-8/12+o(1)}
= R^{-2/3+o(1)}.
That is Theorem 2 part (2c).
The additive C R term. Boundary ports produce O(R) extra
vertices, each contributing at most 1 to the normalized
mean or probability. Compared to the leading R^{4/3} and
R^{-2/3}, that is negligible for large R.
----------------------------------------------------------------
5.5 The H^{2/3} length-squared estimate.
----------------------------------------------------------------
Goal: prove Theorem 2 part (2d).
Reduction to two bonds. Let
S(H) = sum over planar polygons of diameter at most H,
modulo lattice translations,
of x^{|lambda|} |lambda|^2.
The factor |lambda|^2 counts ordered pairs of bonds in
lambda. So S(H) equals the sum over ordered bond pairs of
the two-bond quantity H^poly, estimated in PART 6.
Two-bond estimate (Theorem 3). Uniformly in m, n >= 1 of
any permitted tilted cylinder:
H^poly_{m,n} <= C_eps * n^eps * m^{-4/3} * (n/m)^{1/3}
for every eps > 0.
Good pairs. Every polygon of length |lambda| has at least
|lambda| - 2K vertices far from its sharp turns, for K = 4
fixed. Any bond pair can be replaced by a pair (v_1, v_2)
of good vertices with the same scale, up to a bounded
factor. Count good pairs instead.
Scale identification. For a good pair (v_1, v_2) with
|v_2 - v_1| = d <= H, the polygon sits on a permitted
tilted cylinder of scale (m, n) with m ~ d and
n ~ H - d, up to bounded factors. The selected-pair lemma
(part of Theorem 3 in PART 6) makes this identification
bijective between bond pairs and permitted cylinders, up
to those factors.
Summing over scales. For each scale (m, n) apply the
two-bond bound m^{-4/3} (n/m)^{1/3}. Sum over 1 <= m <=
n <= H with the polynomial pair-count weight, and the
total collapses to
S(H) <= H^{2/3 + eps}.
The two-bond decay in m is what carries the exponent from
2 down to 2/3. In words: critical polygons cannot fill a
disk efficiently. Each extra pair of bonds at scale m
costs m^{-4/3} in the two-bond sum, which dominates the
polynomial growth in the number of such pairs.
Therefore S(H) = O(H^{2/3+eps}) for every eps > 0, which
is Theorem 2 part (2d).
End of PART 5.
================================================================
PART 6. PROVING THEOREM 3 -- TWO MARKED BONDS
================================================================
PART 5 ended with a promise: a two-bond estimate. This part
delivers it. The structure mirrors PART 4 but with two
marked bonds instead of one unmarked cycle. Two new
complications: the two marked bonds break the symmetric
Dirichlet form, and the positive quantity H^poly must be
matched to an explicit signed expression H^flip.
----------------------------------------------------------------
6.1 The two-bond physical identity.
----------------------------------------------------------------
Goal: express the two-bond polygon sum as a signed matrix
element of the finite spin data.
Physical quantity. Let
H^poly(X) = sum over self-avoiding polygons that pass
through both prescribed bonds
of x^{N}.
Here X is the finite site data of the cut, and H^poly is
positive.
Signed counterpart. Run the contour machinery of PART 4 on
the doubled physical sheet with two flips at the two marked
bond midpoints instead of one D_A insertion. The result is
an algebraic expression H^flip(X) whose contour integral
agrees with the physical H^poly in the same regulator
limit. The two endpoint variables U, V are the flip
variables.
Physical identity (Proposition "marked-physical"). At the
physical site list X = (1, ..., 1) with U = V = 1,
H^poly(X) = H^flip(X).
Idea. The signed sum over closed rim states with the two
marked bonds satisfies the same Duminil-Copin-Smirnov
balance identities as the unmarked case, now with two
flips replacing the D_A insertion. Vertex-edge fusion and
arc moves reduce it to the finite spin sum of PART 4. The
regularity of the surface factor S(X)^2 shows the limit at
the physical list is attained; a two-arc phase lemma
cancels the two marked-bond phases. In one line:
H^poly = H^flip at X = physical.
----------------------------------------------------------------
6.2 The generic flip formula.
----------------------------------------------------------------
Goal: an explicit formula for H^flip(X) for generic nonzero
site data.
Setup. Let (A, B) be the two consecutive unmarked lists
with the two marked sites removed. Write U = x_i, V = x_j
for the endpoint variables, and A' = A \ {i},
B' = B \ {j} for the reduced unmarked lists. The reduced
unmarked problem is the finite spin sum of PART 4.
Track the double-flip matrix element through the projectors
and fusion algebra. The result factorizes:
N(X)^2 H^flip(X)
= c U V sum over spins s of q^{-6 Q} C_s
* prod_{a in A' u B'}
F_{s_a}(U, x_a) * F_{s_a}(V, x_a),
where
Q = sum_{A'} s_a = sum_{B'} s_a (charge balance),
e_k(u, v) = v - q^k u, q = e^{2 i alpha},
(F_+, F_0, F_-) = (e_5 e_6, e_3 e_5, e_2 e_3),
c = 2 - sqrt(2).
C_s is the unmarked finite spin summand of (4.3) at the
unmarked lists A', B'.
Reading. H^flip(X) is a product of
- c U V (the two-endpoint scalar),
- q^{-6Q} (the two-flip charge coupling),
- C_s (the unmarked background),
- two per-site factors, one coupling each site to U and
one to V.
Structural facts:
- U and V are independent in the formula. The physical
list will later set U = V = 1.
- Division by N(X)^2, not by any endpoint variable.
- e_k(u, v) vanishes exactly at u = q^k v; those
vanishing patterns are the fusion rules.
----------------------------------------------------------------
6.3 Continuation to physical sites.
----------------------------------------------------------------
Goal: prove H^flip(X) extends holomorphically to the
physical list.
The physical list. In the balanced cut, sites sit at
imaginary heights 0 (in A) and 1 (in B), up to a common
rescaling. Concretely X = (1, ..., 1) of size M = 2L.
Problem. H^flip carries 1/N(X)^2 in front. If N(X)
vanished near the physical list, the continuation would
fail.
Nonvanishing lemma. For a list X of even size 2L with
imaginary site heights in a range of spread at most 1
containing the physical list, satisfying the balancing
condition of 4.3,
N(X) / S(X) = (2/3)^L * E_h * exp( sum_a phi(k_a) ),
where
phi(k) = log( (1 + e^{-2k}) / (1 + e^{-4k}) ),
E_h is a positive real function that tends to 1 as the
spread tends to 0, and S(X) is the explicit Pfaffian
prefactor. In particular, N(X) > 0 on that neighborhood.
Consequence. The explicit polynomial form
N(X)^2 * H^flip(X) makes sense at every site. Since
N(X) != 0 near the physical list, H^flip itself is
holomorphic there. The finite spin sum is computed directly
at X = (1, ..., 1). By 6.1 that value equals H^poly.
So the closed-form expression of 6.2 is valid at the
physical list.
----------------------------------------------------------------
6.4 The marked gas.
----------------------------------------------------------------
Goal: pass the marked algebraic expression through the
contour-to-gas rewriting of 4.4-4.5.
The kink. Start from the unmarked regulator of 4.4 and add
a local "kink" at the endpoint coordinate w_e:
d_T(v) = sum_{ell in Z} [ D_e(v - w_e + ell T)
- D_e(v - w_e - T/2 + ell T) ] - nu.
Reading: two signed step primitives at opposite positions,
minus a normalizing constant. Zero mean, sign-flipping
under T/2-translation, tending to D_e(v - w_e) on compacts
as T -> infinity.
The marked regulator is
C_T^L = E[ e^{i int (DA) chi}
* e^{-2 pi sum_sigma n_sigma ell_sigma / T}
* C^L_{1,T} * C^L_{2,T} ]
/ E[ e^{i int (DA) chi} ],
where C^L_{sigma,T} is the current (4.4) with
phi_sigma(v) -> phi_sigma(v) + d_T(v).
Result (Proposition "marked-regulator-limit"). As T ->
infinity at fixed site count,
C_T^L -> H^flip(X)
computed via the continued algebraic formula of 6.2.
Every surviving residue is exactly one of the finite spin
summands.
Site factors. Particles pick up extra site multiplicities:
P: prod_{w in W_sigma} B_T(x - w)
M, F: 1
O_+: prod_{w in W_sigma} B_T(x - w - i)
O_-: prod_{w in W_sigma} B_T(x - w + i)
U: prod_{w in W_sigma} B_T(x - w + i(3/2+d))^{-1}
L: prod_{w in W_sigma} B_T(x - w - i(3/2+d))^{-1}.
Multiplied over the site multiplicity m. M and F_pair have
no extra factor from this list; their activity comes only
from the local residue.
Gas identity. Let Z_T^L denote the marked gas sum. Then
C_T^L = c(T) Z_T^L
with c(T) the same scalar as in 4.5.
----------------------------------------------------------------
6.5 Stability at displacement d = 1/8.
----------------------------------------------------------------
Goal: prove the absolute stability of 4.6 at d = 1/8,
the displacement the marked problem needs.
Statement (Proposition "stability-eighth"). For the
post-collision gas at d = 1/8, the conclusion of 4.6 holds
with constants uniform in T and theta in a fixed compact
subset.
Idea. The proof of 4.6 was purely a matrix-coercivity
argument on H(k). The same argument works at any fixed d,
as long as the coercivity constant is genuinely positive.
The check at d = 1/8 reduces to explicit rational-function
positivity in the variable u = e^{-k}.
Why d = 1/8 specifically. At the U-L collision, a
"continued H-term" removes the singularity. After the
continuation the H-matrix has explicit rational entries at
each d. Writing a = tanh(Δ) with Δ = k/8, several entries
reduce to
a - 1 = u^2 P_sigma(u) / ((1 - u^2) Q_sigma(u))
for sigma = +/-. The ascending coefficient lists of
P_+, P_-, Q_+, Q_- are all positive integers, so a - 1 is
positive in u for 0 < u < 1. Two further sign conditions
reduce to
(o_0 + b_1 tanh Δ + b_2 tanh^2 Δ) / o_0
with positive coefficients in z = e^{-k/4}. So the
coercivity gap is positive at every k > 0.
Consequence. The absolute stability of 4.6 holds at
d = 1/8, so the marked gas Z_T^L is absolutely convergent
and the trace-extraction lemma of 4.8 applies directly.
----------------------------------------------------------------
6.6 The mixed spectral radius.
----------------------------------------------------------------
Goal: bound the growth of the exterior operator's power
when the block sequence mixes site-perturbed and
site-free segments.
Setup. In the marked problem some blocks carry extra site
multiplicities (central segments), others do not
(exterior segments). Central segments grow at rate u_c;
exterior segments grow at one of several rates, the
largest being r_X^+.
Two-gap lemma (Lemma "trace-extraction" applied to the
product). Power traces split into peripheral-eigenvalue
powers plus exponentially small remainders, uniformly in
the block partition. Every Jordan block is accounted for.
The three rates.
u_c = e^{-pi b / 4} (central blocks)
r_X^+ = exp( b ( -5 pi / 96 + 1 / (6 sqrt 3) ) )
(exterior, largest rate)
r_E = e^{-b pi / 16} (empty, baseline)
The two useful ratios:
log(u_c / r_E) / (k_1 b) = -3 pi / 16,
log(r_X^+ / r_E) / (k_1 b) = pi / 96 + 1 / (6 sqrt 3),
with k_1 = pi / 3. Both are fixed numbers independent of
m, n, theta.
Mixed-radius bound. Suppose a segment pattern has 2J
central blocks and 2 (L - J) exterior blocks. Then for any
fixed eta > 0,
||Z_{4 N b}^L||_1
<= C_eta n^{eta}
* (u_c / r_E)^{2 J}
* (r_X^+ / r_E)^{2 (L - J)}.
Both J and L are O(log n); n^{eta} absorbs the constant
factors. 4.7 ties J to m and L - J to n/m.
----------------------------------------------------------------
6.7 Assembly: Theorem 3.
----------------------------------------------------------------
Goal: prove
H^poly <= C_eps * n^eps * m^{-4/3} * (n/m)^{1/3}.
Start (6.4 + 6.5). Absolute stability of the marked gas
gives
|H^flip(X)| <= C_eps M^{-5/24+eps}
* |lim c(4 N b) Z_{4 N b}^L|.
Mixed-radius bound (6.6). Choose
J = log(m) / (k_1 b) + O(1),
L - J = log(n/m) / (k_1 b) + O(1),
matching the central and exterior segment counts to the
scales m and n/m. Then
||Z_{4 N b}^L||_1 <= C_eta n^eta
* (u_c / r_E)^{2 J}
* (r_X^+ / r_E)^{2 (L - J)}.
Explicit values.
(u_c / r_E)^{2 J}
= m^{-9/8}.
(r_X^+ / r_E)^{2 (L - J)}
= (n/m)^{1/16 + 1/(pi sqrt 3)}.
Combine. With M ~ n:
H^poly <= C_eps n^eps
* n^{-5/24}
* m^{-9/8}
* (n/m)^{1/16 + 1/(pi sqrt 3)}.
Two exponent identities (the algebra of the two ratios):
-5/24 + 2 log(u_c / r_E) / (k_1 b)
= -5/24 - 9/8
= -4/3.
-5/24 + 2 log(r_X^+ / r_E) / (k_1 b)
= -5/24 + 1/16 + 1/(pi sqrt 3)
= 1/(pi sqrt 3) - 7/48
< 1/3.
They combine to the claimed powers:
H^poly <= C_eps n^eps m^{-4/3} (n/m)^{1/3}.
Reading.
- The -4/3 is the polygonal length exponent (the same
R^{4/3} from Theorem 2b).
- The 1/3 balances the sum over scales in 5.5, so that
the two-bond sums converge to the H^{2/3} length-square
bound.
End of PART 6.
================================================================
PART 7. WHAT'S NEW
================================================================
The paper's contributions, ordered by ease of reading.
7.1 The cylinder pressure exponent m^{1/6}.
Many loop models are positivity-dominated (FK percolation,
Ising, Symanzik), so their critical exponents come from
monotonicity arguments. The signed-transfer representation
of this paper has no positivity at all: individual
configurations carry complex or negative weights that can
be large, canceling to a small total. To prove power laws
from such a representation, one needs absolute bounds
strong enough to yield the correct answer. Prior to this
paper, this had not been carried out for the balanced
cylinder at fugacity 2 of the dilute O(n) model. The 1/6
is new.
7.2 The planar nesting exponent 1/12.
Follows from 1/6 by a localization argument (4.11), but
had not been previously available. The 1/12 counts cycle
families, not walk lengths, so it is a genuinely different
quantity from the standard 1/nu = 4/3.
7.3 The chord law.
Mean chord length R^{4/3}; bulk visit probability
R^{-2/3}; total chord mass R^{3/4}. The mean length
matches the conjectured SAW length exponent 4/3, and the
visit probability matches the conjectured R^{-2/3}. The
results are not only for a probability law but for the
exact normalization Z_R^chord.
7.4 The length-square mass H^{2/3}.
A two-bond computation, independent of Theorems 1 and 2.
Any two-bond sum through two specified bonds at distances
m, n is bounded by m^{-4/3} (n/m)^{1/3}, which summed over
scales gives H^{2/3}.
7.5 The method.
The signed transfer / absolute stability / compact
propagator / Jordan-safe trace extraction pipeline is
general-purpose. It should apply to any dilute O(n)
observable once the same absolute stability holds at the
relevant displacement d.
================================================================
PART 8. HONEST LIMITS
================================================================
What the paper does not prove, and where its
simplifications approximate.
8.1 The SAW Flory exponent 3/4.
The mean chord length R^{4/3} matches the conjectured
reciprocal Flory exponent 1/nu = 4/3. But this is a
different object: the normalised mean of first-exit hexagon
chords, not the mean-square end-to-end displacement of a
uniformly chosen length-n walk. The matching exponents are
consistent, not equivalent.
8.2 Uniformity in the exponents, not the constants.
Every R^{a + o(1)} carries a constant that may depend on
the fixed parameters (fugacity, displacement, side length,
cut lengths, counts). The exponents are uniform. Polynomial
prefactors are not.
8.3 No purely logarithmic estimate.
Several intermediate bounds are power bounds with an o(1)
error; they do not give a logarithmic rate. For example,
the arch mass W_R is R^{-1/4+o(1)}, with the o(1) constant
a slowly growing function of R.
8.4 The marked lower bound is not proved.
Theorem 3 is an upper bound on the two-bond sum. A
matching lower bound is not claimed. The applications in
the paper (the H^{2/3} length-square mass) only need the
upper bound.
8.5 Numerical constants.
The seven activities are explicit (sqrt(2)/4 times
exponentials). The stability constants c and C are not.
The Pfaffian scalar is explicit. Optimal choices of
b, J, k_1 are not discussed.
8.6 What is not handled.
- Arbitrary fugacity g. Focus is g = 2 (and complex g near
0 for analytic continuation).
- Arbitrary cycle multiplicities (potentials).
- The limit cases d -> 0 and d -> 1/2 of the displacement
are explicitly not handled uniformly.
- Higher-dimensional analogues are not addressed.
================================================================
PART 9. GLOSSARY
================================================================
Symbols and terms, approximately alphabetical. The
section tags refer to PART numbers in this document.
alpha = pi/8. Turn weight in the parafermionic balance.
arch. A first-exit walk in a half-plane, counted with
weight x^N cos(3 T / 8) after turning.
B(v) = (cosh(pi v / 3) - sqrt(3)/2) / (cosh(pi v / 3) +
sqrt(3)/2). The gas B factor. Zeros at v = i (plus or
minus 1/2 + 6n), poles at v = i (plus or minus 5/2 + 6n).
bad vertex. On a polygon lambda, a vertex within distance
r_K = (K - 1/2) / sqrt(3) of one of its sharp turns. A
polygon has at most 2K bad vertices.
beta = 3/8. Turn-phase ratio s.
c_alpha = 2 sin(alpha). Boundary constant in the arch
mass identity sum_d H(d) = 1 / c_alpha.
chain. A self-avoiding polygon (a closed self-avoiding
walk).
chord. A first-exit walk between two boundary ports.
C_R. The set of first-exit chords in D_R of diameter at
least R / 10, with both boundary ports summed.
critical weight x = (2 + sqrt(2))^{-1/2} = 0.5412...
Multiplies x^N for every length-N walk and polygon.
D_A. The cutting insertion (i h)^{sum_{j in A} s_j} along
the A-cut of the balanced cylinder.
delta = 2 or 3. Fusion index: delta = 2 collapses two
sites to one, delta = 3 to nothing.
D_R. The regular tessellation hexagon of side length R.
discrepancy bound. The constant D of Theorem 3: within
each bin, the count differs from its expected value by at
most D.
E_delta. The smaller fusion image space. E_2 is
3-dimensional (spins -1, 0, +1); E_3 is 1-dimensional.
E_0, I_0. Empty-block operators. E_0 allows all seven
particle types, I_0 allows only M and F_pair. Their
spectral radii are r_E = e^{-b pi / 16} and 1.
E_theta. The homogeneous "all types" block operator at
angle theta.
f = n / (m + n). Aspect ratio of a permitted tilted
cylinder. Bounded by 0.6 in Theorem 3.
F_{s_a}. Per-site factors in the generic flip formula:
F_+ = e_5 e_6, F_0 = e_3 e_5, F_- = e_2 e_3, where
e_k(u, v) = v - q^k u.
F_pair. The fused particle after a U-L collision.
Charge -2, site multiplier 1.
G_m(g) = G_{A,B}(g). The balanced cylinder partition at
fugacity g. With g = 2 cos Theta, its complex
continuation enters the proof.
gamma_r. Small contour around the origin used in 4.4.
gamma_v. The local activity of a gas particle of type v.
Explicit in 4.5.
gamma_{F_pair} = (9 pi^2 / 8) exp( K_diag(0) + K_diag^reg(3i) ).
Positive fused-particle activity.
good vertex. A vertex that is not bad. Every polygon of
length lambda has at least |lambda| - 2K good vertices.
H(d). Half-plane arch weight from a port to a port d
steps to the right. sum_d H(d) = 1 / c_alpha.
H^flip(X). Signed two-flip algebraic expression of 6.2.
Agrees with H^poly at the physical list.
H^poly(X). The positive two-bond sum of Theorem 3.
I_M. Incompatible-pair sum at g = 0. Equals O(log M).
J_delta. Fusion embedding E_delta into V tensor V.
k_1 = pi / 3. Vertical site-rate scale.
K_diag = (K^r)_{sigma sigma}. Diagonal part of the H
kernel.
kappa_T. Gas scalar rate. Limit 2 log(2(2 + sqrt(2))/9).
lambda = e^{-2 i alpha}. The charge-eigenvalue.
M = 2m. Period of the balanced cylinder (even).
mu. Running common charge in the gas.
N(X). Pfaffian scalar. N(1, ..., 1) = (2 p_0 / 3)^{m(2m-1)}
times (2/3)^m times m^{5/48 + o(1)}, with p_0 = 2 + sqrt(2).
omega_v(x). Site multiplier for a particle of type v.
Table in 6.4.
p_*(Theta) = 1/6 - 2 Theta^2 / (3 pi^2). The complex
cylinder exponent.
Pi_p. Spectral projector for twist p. Explicit
Pfaffian det L(T) / det L(1).
q = e^{2 i alpha}. The chiral deformation parameter.
R. Hexagon side length, or the free radius parameter in
W_R.
R(phi). The transfer matrix with five trigonometric
entries (u, d, v, a, b).
R(u, v). Residual pair factor of the gas kernel.
r_E = e^{-b pi / 16}. Spectral radius of E_0.
r_K = (K - 1/2) / sqrt(3). Bad-vertex radius.
r_X^+. The largest exterior rate of 6.6.
S_D(v). The signed visit sum of 5.3.
seam. The twist line where p = i above and p = -i below
the cut.
T. Period (circumference) of the horizontal gas.
Theta. Complex insertion angle, g = 2 cos Theta.
theta. Regulator angle in the complex partition.
u, d, v, a, b. The five trigonometric weights of R(phi).
At phi = alpha: (x, x^2, x^2, x^2, 0).
U, V. The two flip endpoint variables.
u_k = exp( -4 pi b / 9 * (k + theta / pi)^2 ). Eigenvalues
of I_0.
u_c = e^{-pi b / 4}. Central rate of 6.6.
w_e. The endpoint coordinate (marked bond site).
W_R. Arch mass at distance at least R in the half-plane.
Equals R^{-1/4 + o(1)}.
x. See critical weight.
z = e^{-2 i phi}. The transfer-matrix eigenbasis variable.
Z(R). Planar nesting partition around the central face
center in D_R, at fugacity 2. Equals R^{1/12 + o(1)}.
Z_R^chord = sum_{gamma in C_R} x^{N(gamma)}. Chord
normalizer. Equals R^{3/4 + o(1)}.
Z_{m,T}. The exact gas at period T and site count m.
zeta. A peripheral eigenvalue of E_0 (or E').
End of PART 9. End of document.