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.

----------------------------------------------------------------
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
================================================================

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.

================================================================