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-SIGNED CYLINDER PROPAGATION AND MARKED POLYGONS
-ON THE HONEYCOMB LATTICE
-
-A plain-language edition, written to be read front to back.
-
-================================================================
-READ ME FIRST
-================================================================
-
-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.
-
-================================================================
-PART 1. THE STAGE
-================================================================
-
-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.
+The honeycomb lattice H is the graph whose vertices are the centers of the
+triangles of a unit triangular tessellation, with two vertices joined when the
+triangles share a side. Three edges meet at every vertex. Euclidean distance,
+diameter, and all geometric quantities refer to this embedding. Each triangle
+of the tessellation has a center, and every walk below traverses triangle
+centers. A port is the midpoint of a tessellation side on the boundary of a
+region. A first-exit port-to-port walk in a region D is a self-avoiding walk
+inside D with both endpoints on ports of D and no other vertex on a port of D.
+A walk traversing N vertices has weight x^N where x = 1/sqrt(2+sqrt(2)) =
+0.5411961... is the reciprocal of the honeycomb connective constant. Any
+positive weight below is a product of such powers.
+
+The connective constant of the honeycomb lattice equals sqrt(2+sqrt(2)),
+established by
+
+ H. Duminil-Copin and S. Smirnov, The connective constant of the
+ honeycomb lattice equals sqrt(2+sqrt(2)), Annals of Mathematics 175
+ (2012), 1653--1665.
-1.4 Cylinders and separating cycles.
+./signed_cylinder_pdfs/duminil-copin-smirnov-2012-1202.5523-honeycomb-connective.pdf
-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 polygon is a simple closed unoriented cycle on H; its length |lambda| is its
+number of edges. Two polygons are incompatible when they share a vertex; each
+is incompatible with itself. Polygon collections are summed with product
+weights x^{sum |lambda|}.
-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).
+Equilibrium has the following physical interpretation. Nienhuis's Coulomb-gas
+analysis of the O(n) loop model predicted the two-dimensional self-avoiding
+walk exponents, including nu = 3/4. The prediction is recorded in
-1.5 The cylinder partition G_m(g).
+ B. Nienhuis, Exact critical point and critical exponents of O(n) models
+ in two dimensions, Physical Review Letters 49 (1982), 1062--1065,
-Define G_m(g) as the sum, over families of pairwise disjoint
-separating cycles, of
+and in the review
- g^k * x^{|lambda_1| + |lambda_2| + ... + |lambda_k|}
+ B. Nienhuis, Critical and multicritical O(n) models, Physica A 163
+ (1990), 152--157.
-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 exponent nu = 3/4 corresponds to reciprocal 1/nu = 4/3, which appears
+below as the mean-first-exit-chord exponent and as the leading decay of the
+marked two-bond sum. The empirical vanishing of critical bridges was proved
+by Beaton, Bousquet-Melou, de Gier, Duminil-Copin and Guttmann in
-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.
+ N. R. Beaton, M. Bousquet-Melou, J. de Gier, H. Duminil-Copin and A. J.
+ Guttmann, The critical fugacity for surface adsorption of self-avoiding
+ walks on the honeycomb lattice is 1+sqrt(2), Communications in
+ Mathematical Physics 326 (2014), 727--754.
-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).
-
-================================================================
-PART 2. THE THREE THEOREMS
-================================================================
-
-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.
-
-----------------------------------------------------------------
-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.
-
-----------------------------------------------------------------
-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.
-
-----------------------------------------------------------------
-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.
-
-================================================================
-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.
-
-================================================================
-
-================================================================
-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.
-
-----------------------------------------------------------------
-4.1 The oriented transfer and the height limit.
-----------------------------------------------------------------
-
-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.
-
-----------------------------------------------------------------
-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,
+./signed_cylinder_pdfs/beaton-bousquet-melou-de-gier-duminil-copin-guttmann-2014-1406.3163.pdf
- Pi_p = det L(T) / det L(1),
+For honeycomb bridges Glazman and Manolescu obtained a logarithmic
+subsequence bound, and for columnwise rhombic half-plane tilings they proved
+invariance of boundary two-point functions; see
-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.
+ A. Glazman and I. Manolescu, Self-avoiding walk on Z^2 with Yang-Baxter
+ weights, Annales de l'Institut Henri Poincare B 56 (2020), 2281--2300.
-The denominator is a Pfaffian:
+Krachun and Panagiotis obtained a polynomial upper bound on the critical bridge
+partition and used it to prove quantitative sub-ballisticity for uniform
+honeycomb walks; see
- det L(1) = (Pf S(Y))^2,
+ D. Krachun and C. Panagiotis, Quantitative sub-ballisticity of
+ self-avoiding walk on the hexagonal lattice, Annals of Probability 54
+ (2026), 1109--1125.
-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_+.
+The local mechanism on which the present argument depends is the parafermionic
+cancellation of Duminil-Copin and Smirnov, extended to rhombic Yang-Baxter
+weights by Glazman and Manolescu and by Ikhlef and Cardy:
-Inserting the projectors and applying fusion gives the
-finite spin sum formula: it expresses N(X)^2 G_{A,B}(h+h^{-1})
-as
+ Y. Ikhlef and J. Cardy, Discretely holomorphic parafermions and integrable
+ loop models, Journal of Physics A 42 (2009), 102001.
- 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.
+Three observables are treated. The first is a cylinder partition with two
+marked points. Fix an even integer M = 2m, identify horizontal tessellation
+sites modulo M, and mark the face centers O = (0,0) and O' = (m,0) on the cut.
+The cut sites are split into consecutive lists A (from O to O') and B (from
+O' back to O), each of size m. The cylinder is infinite vertically and is
+defined by the increasing height limit of finite cylinders with both rims
+closed (every port on either rim unused). A separating cycle is a simple
+closed cycle of H whose complement places O and O' in different components.
+Let G_m(g) be the sum over families of pairwise disjoint separating cycles of
-Structural facts about the sum:
+ prod_{lambda in family} g x^{|lambda|},
- - 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.
+the empty family contributing 1. Because a separating cycle crosses the cut
+between the marked points, G_m(g) has degree at most 2m in g.
-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.
+The second observable is the chord law inside a regular tessellation hexagon
+D_R of side R. Let Z(R) be the sum over collections of disjoint polygons
+inside D_R that surround the central face, with weight 2 x^{|lambda|} per
+polygon and empty collection contributing 1. Let C_R be the set of first-exit
+chords of D_R of diameter at least R/10, where a chord is a first-exit walk
+whose endpoints are boundary ports of D_R, the sum runs over every starting
+port and every possible first-exit port, and the direction from start to exit
+is retained. Define
+
+ Z_R^chord = sum_{gamma in C_R} x^{N(gamma)},
+ P_R^chord(gamma) = x^{N(gamma)} / Z_R^chord.
+
+The chord length N(gamma) is the number of vertices of gamma.
+
+The third observable is the two-bond sum for planar polygons. For H > 0, put
+
+ S(H) = sum_{diam lambda <= H} x^{|lambda|} |lambda|^2,
+
+summed over polygons modulo lattice translations. The squared length factor
+|lambda|^2 counts ordered pairs of bonds inside lambda; the sum over such
+pairs decomposes S(H) into a sum over two-bond configurations, whose control
+is the content of the third theorem.
+
+Through all statements, F(R) = R^{a+o(1)} means that log F(R) / log R
+converges to a, an upper bound F(R) <= R^{a+o(1)} means: for every eps > 0
+there are C_eps and R_eps such that F(R) <= C_eps R^{a+eps} for all
+R >= R_eps, and a lower bound is analogous. A bound is uniform in some data
+when the constants C_eps and thresholds R_eps do not depend on that data.
+
+Theorem (cylinder pressure and planar nesting). As m -> infinity through even
+positive integers, the balanced cylinder partition with two marks satisfies
-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
+ G_m(2) = m^{1/6+o(1)},
+
+ -(d^2/dg^2) log G_m(g) at g = 0 is O(log m),
+
+the derivative being the total weight of ordered pairs of incompatible
+separating cycles (pairs sharing at least one vertex, including each cycle
+with itself), and locally uniformly for complex Theta with |Re Theta| < pi,
+
+ |G_m(2 cos Theta)| <= m^{Re(1/6 - 2 Theta^2/(3 pi^2)) + o(1)}.
+
+The planar nesting partition satisfies Z(R) = R^{1/12+o(1)} as R -> infinity
+through integers.
+
+Theorem (finite chords and marked polygons). For the chord law of D_R,
+
+ Z_R^chord = R^{3/4+o(1)},
+
+ E_R^chord N = R^{4/3+o(1)},
+
+and uniformly for honeycomb vertices v at distance at most R/10 from the
+central face,
+
+ P_R^chord(v lies on gamma) = R^{-2/3+o(1)}.
+
+Moreover,
+
+ S(H) <= H^{2/3+o(1)}.
+
+Theorem (unequal marked intervals). On a physical tilted cylinder whose
+unmarked lists have m+1 and n+1 sites, with n >= m and M = m+n+2 even, with
+unmarked logarithmic heights -f and 1-f where 0 <= f <= 0.6, and with site
+counts differing from m(1-f), mf and n(1-f), nf by at most a fixed constant,
+let H^poly be the total critical weight of a single unoriented polygon
+crossing two horizontal-type marked bonds. Then uniformly over the lists and
+orders, for every eps > 0 there are C_eps and m_eps such that
+
+ H^poly <= C_eps n^{eps} m^{-4/3} (n/m)^{1/3}
+
+for all n >= m >= m_eps.
+
+The remainder of the paper proves these statements in an interleaving that
+mirrors their mechanism. The boundary balance and local transfer identities
+are established first, since they seed the local cancellation and the finitude
+of vacuum limits. The half-plane arch estimate then controls boundary mass.
+The cylinder theorems proceed through an oriented transfer, Yang-Baxter and
+fusion identities, a finite spin sum, contour and bosonization rewritings, an
+absolute stability estimate on the resulting signed gas, and a compact
+propagator that isolates the physical power. The chord and visit corollaries
+follow by combining the arch estimate, a boundary visit identity, and the
+planar nesting exponent. The two-bond estimate is proved independently, with
+an analogous signed-transfer pipeline but with two marked bonds instead of one
+unmarked cycle.
+
+Throughout, s = 3/8 and alpha = pi/8, so 2 x cos(s pi / 3) = 1. The seed is
+the parafermionic turn balance of Duminil-Copin and Smirnov.
+
+Lemma (convex balance). Let D be any finite convex tessellation polygon, and
+let a be a port on the boundary of D. The signed sum over first-exit port-to-
+port walks gamma of D from a, weighted by x^{N(gamma)} e^{i s T(gamma)}, where
+T(gamma) is the total turning of gamma in radians and orientation is counted
+with sign, equals one.
+
+Proof. Start with the nonbacktracking tree of walk continuations from a,
+with an outgoing factor x e^{i s pi/3} or x e^{-i s pi/3} at each vertex
+depending on whether the walk turns left or right. The two outgoing factors
+sum to 2 x cos(s pi / 3) = 1 at every interior vertex. The tree stops the
+first time an already-visited vertex is about to be entered. Every leaf of
+this tree is one of the following: a boundary port (that is, a first-exit
+walk, which is the quantity being summed), or a first self-collision loop,
+closed by the incoming stem.
+
+A first self-collision closes a simple loop. In the flat annular region
+formed by the loop together with the stem, the stem approaches the loop from
+outside. The loop's turn contribution is either +5pi/3 or -5pi/3 (the loop
+itself has turn +-2pi, but one turn of pi/3 at the closure corner has been
+dropped because only two of the three branches at that interior vertex are
+used), and the stem's turn onto the loop is -+pi/3. The two orientations of
+the loop are summed with weight e^{i s(+-5pi/3 -+ pi/3)}. The total turn per
+orientation is +-4pi/3, so the combined phase contribution at the self-
+collision is 2 cos(4 pi s / 3) = 2 cos(pi / 2) = 0. Every self-collision
+therefore cancels. Only first-exit walks remain, and the total signed sum
+equals the same factor that the interior vertices put at each turn, which is
+one.
+
+For a first-exit walk to a port b, the total turning T(gamma) equals Delta -
+pi, where Delta is the counterclockwise boundary tangent turn accumulated
+from a to b along the boundary, together with the two extra quarter turns
+needed to close the polygon. Each boundary traversal below contributes
+cos(s T) >= sin(alpha) > 0 because Delta is a boundary turn plus a fixed
+compensation. Taking real parts of the balance and using the positivity gives
+the finiteness of the total positive weight of first-exit walks. QED
+
+Let H(d) be the total positive weight x^N of first-exit walks on the half-
+plane above a horizontal tessellation line, from a fixed boundary port to the
+port d steps to the right. Reflection in the vertical axis gives the same
+total to the left. Taking real parts of the convex balance in the half-plane
+and using the two equal sides, gives
+
+ 2 sin(alpha) sum_{d>0} H(d) <= 1,
+
+with strict equality in the limit and no contribution from d = 0 (a first-exit
+arch has positive length). In particular sum_{d>0} H(d) <= 1/(2 sin alpha) <
+3/2, so the sum converges.
+
+The following transfer-matrix framework realizes the balance identically as a
+matrix-element equation. Consider a cylinder obtained by identifying modulo M
+steps horizontally, with rows tiled by M rhombi of basis 1 and e^{i theta},
+with theta potentially varying by row. Inside each rhombus, connections
+(possibly empty) pair up bond midpoints on distinct sides, using each midpoint
+at most once, noncrossingly. The empty weight is 1, a straight connection
+between opposite sides weighs v, single turns of magnitudes theta and
+pi - theta weigh u and d respectively (where port directions are normal to
+sides), and two simultaneous turns weigh a for the pairing left-down and
+up-right, or b for left-up and right-down. Write phi = 3 theta / 8 and let
+s_k = sin(phi + k alpha), h = sin(2 alpha). The trigonometric SAW weights are
+
+ (u, d, v, a, b) = (h s_5, h s_0, s_0 s_5, s_5 s_6, -s_0 s_7) / (s_2 s_3).
+
+At phi = alpha these are (x, x^2, x^2, x^2, 0), exactly the honeycomb weights
+after splitting each rhombus into the two equilateral triangles.
+
+For a finite stack, close both rims. The partition Z of one unoriented
+noncontractible polygon is the linear coefficient in a fugacity n of the full
+loop sum assigning weight zero to contractible loops and weight n per
+noncontractible loop. For real angles 0 < theta < pi, orient each loop with a
+factor exp(-i T / 4) and twist p to the signed number of seam crossings (east
+positive); summing over the two orientations produces the loop weights n =
+p + p^{-1}. This is because an embedded loop in the flat cylinder is either
+contractible with turn +-2 pi, or has winding +-1 and turn zero. The turning
+identities then extend to the rhombic model algebraically.
+
+The transfer matrix. On each horizontal-flow and descending-flow bond put a
+spin in {0, +1, -1}. The oriented matrix acts with row indices (left, up) and
+column indices (right, down), with positive signs meaning left-to-right or
+up-to-down on the corresponding bond. After removing the telescoping phase
+e^{i theta (left - right) / 4}, the resulting R(phi) is block-diagonal with
+respect to the total sum s_left + s_up = s_right + s_down = J. Writing
+lambda = e^{-2 i alpha}, the blocks are
+
+ (s', s'), s' = +-1 : [a]
+
+ (s', 0), (0, s') : [[v, u], [u, v]]
+
+ (1, -1), (0, 0), (-1, 1) : a 3 x 3 block
+ [[b, d lambda, b lambda^2 + a],
+ [d lambda^{-1}, 1, d lambda],
+ [b lambda^{-2} + a, d lambda^{-1}, b]].
+
+The turns connecting left to down have magnitude theta (weights u, d, a, b as
+above); those connecting left to up have magnitude pi - theta (weights v, u,
+d, b, a with the substitutions s_j -> s_{j+4} of the trig functions).
+
+Exchange. Let V = C^3 with basis |0>, |+>, |->, and let subscripts index
+ordered tensor factors. On a copy V_i let s_i act by s_i |k> = k |k>. Write
+P_{ij} for interchange of factors i, j, and put Delta_R(psi) = sin(psi + 2
+alpha) sin(psi + 3 alpha).
+
+Lemma (local exchange and fusion). The entries of R(psi) are meromorphic,
+with possible poles at Delta_R(psi) = 0, and the following are identities of
+meromorphic matrix functions:
+
+ R_{12}(t) R_{13}(t + phi) R_{23}(phi) = R_{23}(phi) R_{13}(t + phi) R_{12}(t),
+
+ R_{12}(phi) R_{21}(-phi) = I.
+
+In particular R_{12}(0) = P_{12}.
+
+Proof. The unitarity R_{12}(phi) R_{21}(-phi) = I follows by block multipli-
+cation, using lambda^4 = -1 and the elementary relation aa' = 1, ab' + ba' +
+dd' = 0, ad' + d = a'd + d' = 0, vv' + uu' = 1, vu' + uv' = 0, where primes
+denote the same quantity at -phi. Each displayed scalar identity reduces to
+sin_j' = sin_{8-j} and product-to-sum.
+
+To prove the Yang-Baxter exchange at the three special values t = 0, 2 alpha,
+3 alpha, it suffices to check the factorizations R_{12}(0) = P_{12},
+R_{12}(2 alpha) = J_2 J_2^*, R_{12}(3 alpha) = J_3 J_3^*, together with the
+three-space identity in each block; the block calculations are routine. For
+general t the exchange identity is proved by noting that, upon multiplying
+the exchange difference by the product of the three Delta_R factors appearing
+(t, t + phi, phi), every entry becomes a polynomial in e^{2 i t} of degree at
+most 4. The six values 0, 2 alpha, 3 alpha, -phi, 2 alpha - phi, 3 alpha - phi
+are distinct modulo pi for generic real phi, so this polynomial of degree at
+most 4 vanishes if it vanishes at these six values. Each of the six reduces
+by the swap symmetry R(psi + pi) = Z_1 R(psi) Z_1 where Z |k> = (-1)^{k^2}
+|k>, to one of the factorizations at 0, 2 alpha, 3 alpha. Continuation in the
+two parameters proves the meromorphic identity. QED
+
+The exchange identity has a consequence about the family of rows: rows of any
+quantity whose entries are R-matrices with a common twist have a simultaneous
+eigenbasis, and their eigenvalues depend only on the row's rapidity. Fusion
+extends this to admitting a row to be collapsed.
+
+Fusion. Two adjacent rows can collapse when their rapidities differ by a
+quantized amount. Set E_2 = V and E_3 = C, and define injective maps
+J_delta : E_delta -> V tensor V by
+
+ delta = 2:
+ J_2 |0> = |00> + x sum_{k = +-1} lambda^k |k, -k>
+ J_2 |k> = x (|k, 0> + |0, k>) for k = +-1
+
+ delta = 3:
+ J_3 (1) = sum_{k = 0, +-1} lambda^k |k, -k>.
+
+On E_2 put s_f |k> = k|k>, and on E_3 put s_f = 0.
+
+Lemma (fusion factorization). For delta = 2, 3 the identity
+
+ R_{12}(delta alpha) = J_delta J_delta^*
+
+holds in V tensor V, and for any extra factor V_k the meromorphic site
+intertwining holds:
+
+ R_{23}(phi) R_{13}(phi + delta alpha) (J_delta tensor I)
+ = (J_delta tensor I) R_f(phi)
+
+ R_{a1}(psi) R_{a2}(psi + delta alpha) (I_a tensor J_delta)
+ = (I_a tensor J_delta) R_f(psi + [alpha for delta = 2 else 0])
+
+with R_f(phi) = R(phi + alpha) for delta = 2 and R_f = I for delta = 3.
+
+Proof. At the three values psi = 0, 2 alpha, 3 alpha the five weights are,
+respectively,
+
+ psi = 0 : (1, 0, 0, 1, 0)
+ psi = 2 alpha : (x^2, x, x^2, 0, x^2)
+ psi = 3 alpha : (0, 1, 0, 0, 1).
+
+The R(psi) block decompositions at these values produce the depicted images
+directly. For instance, the charge-zero column of J_2 is (x lambda, 1, x
+lambda^{-1})^T, and the charge +-1 columns are x (1, 1)^T; substituting into
+J_2 J_2^* reproduces R_{12}(2 alpha). The factorizations are not isometries:
+J_2^* J_2 = diag(3 - sqrt 2, 2 - sqrt 2, 2 - sqrt 2) in the standard basis,
+and J_3^* J_3 = 3.
+
+The three-space intertwining is verified by explicit block multiplication in
+each charge sector. The scalar identities needed reduce, after cancelling
+common factors, to a finite list of sine product-to-sum identities together
+with lambda^2 + lambda^{-2} = 0; the block-by-block check is direct. The
+row-space formula j:site-fusion is the transpose-in-spaces image of the
+two-space version, obtained by conjugating the row intertwining and exchanging
+factors 1 and 2, using the invariance of J_delta under complex conjugation in
+real rapidities, and then continuing meromorphically. QED
+
+Lemma (restriction to fusion image, and quotient at fusion). Fix delta in
+{2, 3} and put W_delta = im J_delta subset V_1 tensor V_2. For an extra factor
+V_k set F_{delta, k}(psi) = R_{2k}(psi) R_{1k}(psi + delta alpha). At regular
+parameters the subspace W_delta tensor V_k is invariant; transported through
+J_delta tensor I the restriction reproduces R_f(psi). At psi = 0 the full
+image of F_{delta, k}(0) lies in W_delta tensor V_k, so the induced operator
+on ((V_1 tensor V_2) / W_delta) tensor V_k is zero at psi = 0 and holomorphic
+near psi = 0 (being a polynomial in the local parameter divided by Delta_R
+factors that do not vanish there). The twist restricts by
+p^{s_1 + s_2} J_2 = J_2 p^{s_f} and p^{s_1 + s_2} J_3 = J_3 for every p != 0.
+
+Proof. The invariance and the action on the image are the site intertwining
+of the previous lemma, transported through the identification E_delta -> W_delta
+supplied by J_delta. At the fusion point the factorization supplies the
+stronger inclusion
+
+ F_{delta, k}(0) = P_{2k} R_{1k}(delta alpha)
+ = P_{2k} J_{delta, 1k} J_{delta, 1k}^*,
+
+ im F_{delta, k}(0) subset W_delta tensor V_k,
+
+where J_{delta, 1k} is the embedding applied on factors 1 and k. Indeed
+P_{2k} carries im J_{delta, 1k} onto im J_{delta, 12} = W_delta. The quotient
+operator is holomorphic at psi = 0 because the local parameters 0, delta alpha
+are regular; its vanishing at psi = 0 then factors as (z - 1) times a
+holomorphic function in the local coordinate z = e^{2 i psi}. The twist
+identities follow because every term of J_2 |s_f> has total spin s_f, and
+every term of J_3 has total spin zero. QED
+
+Fourier kernel. The finite-vacuum normalization and the boundary moment
+problem both use the same elementary transform.
+
+Lemma (Fourier kernel). For 0 < beta < pi and |Im y| < pi - beta,
+
+ int_R e^{i k y} (sinh(beta k) / sinh(pi k)) dk
+ = sin(beta) / (cosh y + cos beta).
+
+The integral and all parameter derivatives converge locally uniformly on
+the indicated strip.
+
+Proof. For real y > 0 close the contour by rectangles in the upper half-plane
+with upper edges at half-integer heights. The ratio of hyperbolic sines decays
+exponentially on the vertical edges, and the factor e^{i k y} controls the
+upper edge as its height tends to infinity. The residues at k = i n (n >= 1)
+give
+
+ 2 sum_{n >= 1} (-1)^{n + 1} sin(n beta) e^{-n y}
+ = sin(beta) / (cosh y + cos beta).
- |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
+The geometric-series evaluation proves the last equality. Evenness extends it
+to negative y, and dominated convergence covers y = 0. Exponential decay in
+|k| proves local uniform convergence and parameter differentiation on smaller
+strips. Analytic continuation in y gives the asserted domain. QED
+
+The transfer formalism, the convex balance, and the fusion identities seed
+everything that follows. Their combination produces a signed-transfer
+representation of both the cylinder partition and the two-bond sum. The
+remaining work is to make the representation's individual signed terms
+summable and to read off the exponents from the absolute sum.
+
+The half-plane arch estimate. Fix a port on the boundary of a horizontal
+tessellation half-plane. Let W_R be the total critical weight x^N of first-
+exit same-boundary arches reaching Euclidean distance at least R from that
+port, and for d > 0 let H(d) be the total weight of first-exit arches from
+the fixed port to the port d boundary steps to its right.
+
+Proposition (boundary arch mass). W_R = R^{-1/4+o(1)}, and
+
+ 2 sin(pi/8) sum_{d>0} H(d) = 1,
+ sum_{d >= n} H(d) <= n^{-1/4+o(1)}.
+
+Proof.
+
+The fused cylinder increment. On a finite cylinder of circumference M with
+closed rims, let Z be the partition of exactly one unoriented noncontractible
+polygon. Write T_p(phi) for the oriented row transfer matrix with row
+parameter phi, and put alpha = pi/8. Row matrices with the same twist commute:
+multiplying the local exchange identity across the sites and tracing over the
+two auxiliary spaces, the twist p^{s_1 + s_2} commutes with the exchange
+matrix. Then they commute wherever that matrix is invertible, and identically
+by continuation.
+
+Set U = im J subset V_1 tensor V_2 for the embedding at 2 alpha, and at site
+j put X_j(phi) = R_{2j}(phi) R_{1j}(phi + 2 alpha). Locally this preserves
+U tensor V_j and induces R(phi + alpha) through the isomorphism J; at phi = 0,
+
+ X_j(0) = P_{2j} R_{1j}(2 alpha), im X_j(0) subset U tensor V_j,
+
+because R_{1j}(2 alpha) = J_{1j} J_{1j}^* and the swap carries embedded
+spaces 1, j to 1, 2. The operator induced on the quotient by U tensor V_j is
+O(phi), so its product over M sites is O(phi^M). Taking the auxiliary trace
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.
+ T_p(phi) T_p(phi + 2 alpha) = T_p(phi + alpha) + O(phi^M), (*)
-================================================================
-PART 7. WHAT'S NEW
-================================================================
+locally holomorphic in (phi, p) near (0, i). Extract the coefficient of n =
+p + p^{-1}: at p = i, dn/dp = 2, which preserves the order of vanishing.
-The paper's contributions, ordered by ease of reading.
+Now add one new outer row on L honeycomb rows with all parameters alpha and
+subtract the partition without the new row. Let H_M(d) be the weight of half-
+plane arches from a fixed port to the port d steps right, disjoint from all
+their nontrivial horizontal translates by M. For fixed M these converge as
+L -> infinity, since dropping mutual disjointness bounds each by products of
+finitely many H(d) and the coarse summability from the convex balance applies.
+The tile weights satisfy u d = c_alpha v (1 - v) with c_alpha = 2 sin alpha.
+A k-interval term has at most M - 2k straight tiles, so the limiting increment
+is a polynomial F(v) of degree at most M, and
-7.1 The cylinder pressure exponent m^{1/6}.
+ F(v) = v^M + M c_alpha v (1 - v) sum_{d = 1}^{M-1} v^{M - d - 1} H_M(d)
+ + (1 - v)^2 Q_M(v),
-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.
+ F(1) = 1,
+ F'_v(1) = M (1 - c_alpha sum_{d = 1}^{M-1} H_M(d)),
-7.2 The planar nesting exponent 1/12.
+for a polynomial Q_M. Inclusion--exclusion for two added rows, and domination
+by positive honeycomb weights, pass (*) to the limit. With y = i(2 phi - 3
+alpha) and v = (cosh y - sin alpha) / (cosh y + cos alpha), the resulting
+identity near y = -3 i alpha is
-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.
+ f(y) + f(y + 4 i alpha) - f(y + 2 i alpha) = 1 + O((y + 3 i alpha)^M),
-7.3 The chord law.
+where f(y) = 1 - F(v(y)).
-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.
+Positive moment representation. Index columns j = 0, ..., M - 1, and set
-7.4 The length-square mass H^{2/3}.
+ A_j(k) = partial_alpha^j sinh(alpha k),
+ B_j(k) = partial_alpha^j cosh(alpha k),
+ w(k) = 2(1 + 4 sinh^2(alpha k)) / sinh(pi k),
+ w_0(k) = k w(k).
-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}.
+Since F(1) = 1, the function f(y) is a linear combination of (cosh y + cos
+alpha)^{-r}, 1 <= r <= M. The first M parameter derivatives of sin alpha /
+(cosh y + cos alpha) = int_R e^{i k y} sinh(alpha k) / sinh(pi k) dk span
+these powers: the derivative of order j has a nonzero coefficient of the
+denominator power j + 1. The Fourier kernel lemma gives a vector c with
-7.5 The method.
+ f(y) = int_R e^{i k y} A(k)^T c / sinh(pi k) dk,
-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.
+and the integral and required derivatives converge locally uniformly in
+|Im y| < pi - alpha. At y = -3 i alpha, splitting positive and negative k gives
+the moment matrix equation
-================================================================
-PART 8. HONEST LIMITS
-================================================================
+ D c = B(0), D = int_0^infty B(k) A(k)^T w(k) dk.
-What the paper does not prove, and where its
-simplifications approximate.
+D is nonsingular. The ordered determinant expansion of det D is an integral
+over 0 < k_1 < ... < k_M of a product of two determinants times positive
+weights. Both factors are strictly positive: expanding sinh or cosh in
+positive power series gives each via a positive generalized Vandermonde, and
+the sign of the second follows by separating the positive k_i in scale. Then
+det D = int over configurations of (positive factor) times Delta(k)^2, hence
+> 0.
-8.1 The SAW Flory exponent 3/4.
+Boundary residue. Set K(z) = A(z)^T D^{-1} B(0). For real y -> +infinity,
+the residue at k = i of the Fourier formula for f gives -2 i K(i) e^{-y} as
+its first term. On the polynomial side,
-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.
+ 1 - v = 2(cos alpha + sin alpha) e^{-y} + O(e^{-2 y}),
-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.
+so
-good vertex. A vertex that is not bad. Every polygon of
-length lambda has at least |lambda| - 2K good vertices.
+ F'_v(1) = -i K(i) / (cos alpha + sin alpha).
-H(d). Half-plane arch weight from a port to a port d
-steps to the right. sum_d H(d) = 1 / c_alpha.
+Orbital form. The positive density
-H^flip(X). Signed two-flip algebraic expression of 6.2.
-Agrees with H^poly at the physical list.
+ det[A(k_1), ..., A(k_M)] det[B(k_1), ..., B(k_M)] prod_i w(k_i)
-H^poly(X). The positive two-bond sum of Theorem 3.
+on 0 < k_1 < ... < k_M is realized by the Harish-Chandra--Itzykson--Zuber
+integral: for D_k = diag(k_1, ..., k_M, -k_1, ..., -k_M) and Q = U diag(alpha
+I, -alpha I) U^* with U Haar on U(2M), put I(k) = E exp(tr(D_k Q)). Then the
+density is proportional to
-I_M. Incompatible-pair sum at g = 0. Equals O(log M).
+ Delta(k^2)^2 prod_i w_0(k_i) I(k),
-J_delta. Fusion embedding E_delta into V tensor V.
+which is a positive probability density. The derivation from the HCIZ formula
+follows by compressing U D_u U^* to a (2M - 1)-dimensional minor; the ordered
+eigenvalues interlace, the last resolvent residue gives uniform simplex
+weights, and the resulting Jacobian is Delta(l) / Delta(u). Induction with
+test parameters z_i - z_d and column subtraction produce the paired
+determinantal formula. Specializing to paired lists (k_1, ..., k_M, -k_1,
+..., -k_M) and (u_1, ..., u_M, -u_1, ..., -u_M), using the paired Vandermonde
-k_1 = pi / 3. Vertical site-rate scale.
+ |Delta(+-k)| = 2^M (prod_i k_i) Delta(k^2)^2,
-K_diag = (K^r)_{sigma sigma}. Diagonal part of the H
-kernel.
+and splitting the exponential determinant into sinh and cosh blocks gives the
+form. The orbital density extends continuously when one point approaches
+zero, and it is integrable:
-kappa_T. Gas scalar rate. Limit 2 log(2(2 + sqrt(2))/9).
+ w_0(k) ~ 4 k e^{-6 alpha k} (k -> infinity), w_0(0) = 2 / pi.
-lambda = e^{-2 i alpha}. The charge-eigenvalue.
+Pinning. Let rho(0) be the one-point intensity at zero of the orbital law.
+Pinning one point there leaves N = M - 1 positive points with density pro-
+portional to Delta(k^2)^2 prod_{j = 1}^N k_j^4 w_0(k_j) I(k_1, ..., k_N, 0),
+where the orbital integral still uses the 2M-dimensional orbit with one zero
+pair in D_k. For its expectation E_pin,
-M = 2m. Period of the balanced cylinder (even).
+ sin alpha / alpha * rho(0)/w_0(0) * E_pin prod_j (1 + k_j^{-2})
+ <= -i K(i)
+ <= rho(0)/w_0(0) * E_pin prod_j (1 + k_j^{-2}).
-mu. Running common charge in the gas.
+The pinning proof compares determinants with one evaluated at the missing
+zero, using a Bessel function identity
-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).
+ (sinh(k_j t)/f_z(t))' = ((k_j^2 - z^2) / f_z(t)^2)
+ int_0^t f_z(v) sinh(k_j v) dv
-omega_v(x). Site multiplier for a particle of type v.
-Table in 6.4.
+for f_z(u) = sinh(z u)/z, so that ratios of determinants reduce to positive
+weighted averages. The averaging weights satisfy the elementary bounds
+g(u_M) <= R <= g(u_1)/g(u_M) for g(t) = sin t / t, giving the displayed
+two-sided pinning bound.
-p_*(Theta) = 1/6 - 2 Theta^2 / (3 pi^2). The complex
-cylinder exponent.
+Orbital derivatives and stochastic comparison. For either orbital factor,
+there are q_j >= 0 with partial_{k_j} log I = 2 q_j and
-Pi_p. Spectral projector for twist p. Explicit
-Pfaffian det L(T) / det L(1).
+ 0 <= q_j <= min{alpha, alpha^2 k_j, alpha^2 / sum_{ell} (k_j + k_ell)^{-1}}.
-q = e^{2 i alpha}. The chiral deformation parameter.
+Derivation: set q_j = <Q_{+j, +j}> under the tilting exp(tr(D_k Q)), exchange
+all paired positions and send Q to -Q to get <Q_{-j, -j}> = -q_j. Convexity in
+k_j from the log-moment-generating structure gives q_j >= 0, and ||Q|| = alpha
+gives q_j <= alpha. Infinitesimal conjugation gives
+<Q_{aa} - Q_{bb}> = (D_k{aa} - D_k{bb}) <|Q_{ab}|^2>; for a = +j, b = -ell
+this becomes <|Q_{+j, -ell}|^2> = (q_j + q_ell)/(k_j + k_ell). Summing over
+all ell and using sum_b <|Q_{+j, b}|^2> = alpha^2 (from Q^2 = alpha^2 I) and
+dropping q_ell >= 0 gives the third bound.
-R. Hexagon side length, or the free radius parameter in
-W_R.
+The orbital density's log-derivative relative to the pure density with weight
+k^e w_0(k) e^{2 alpha k} is 2 q_i - 2 alpha <= 0; positive association of the
+underlying pure law (Vandermonde determinant squared) then gives
-R(phi). The transfer matrix with five trigonometric
-entries (u, d, v, a, b).
+ O_N^{(e)} <=_st P_N[k^e w_0(k) e^{2 alpha k}], e = 0, 4,
-R(u, v). Residual pair factor of the gas kernel.
+and the first j such points are stochastically below P_j[k^e w_0(k) e^{2 alpha
+k}]. The logarithm of w_0 has derivative D_0(k) satisfying
-r_E = e^{-b pi / 16}. Spectral radius of E_0.
+ -6 alpha < D_0(k) < 0,
+ D_0(k) -> -6 alpha (k -> infinity),
+ D_0(k) = -(40/3) alpha^2 k + O(k^3) (k -> 0).
-r_K = (K - 1/2) / sqrt(3). Bad-vertex radius.
+Consequently, with m = floor(N^{1 - epsilon}), on an event E of probability
+1 - O(e^{-c m}) the small-index coordinates satisfy
-r_X^+. The largest exterior rate of 6.6.
+ sup_{i <= m, k in E} q_i <= C / (epsilon log N) = o(1).
-S_D(v). The signed visit sum of 5.3.
+Gamma comparison and evaluation. For ell, beta > 0 let W_{ell, beta}(k) =
+|Gamma(ell + i beta k)|^2 and b = pi beta. For every fixed b_+ > 6 alpha there
+is ell_+ > 0 such that with beta_+ = b_+/pi,
-seam. The twist line where p = i above and p = -i below
-the cut.
+ P_N[k^e W_{ell_+, beta_+}] <=_st O_N^{(e)} for all N, e = 0, 4.
-T. Period (circumference) of the horizontal gas.
+For fixed 0 < b_- < 6 alpha and 0 < epsilon < 1 there is ell_- > 0 such that,
+with beta_- = b_-/pi and for sufficiently large N,
-Theta. Complex insertion angle, g = 2 cos Theta.
+ L_{O_N^{(e)}}(k_{<= m} | E, k_{> m}) <=_st P_m[k^e W_{ell_-, beta_-}]
-theta. Regulator angle in the complex partition.
+almost everywhere on later configurations compatible with E. The Gamma
+derivative
-u, d, v, a, b. The five trigonometric weights of R(phi).
-At phi = alpha: (x, x^2, x^2, x^2, 0).
+ G_{ell, beta}(k) = -2 beta^2 k sum_{n >= 0} 1/((n + ell)^2 + beta^2 k^2)
-U, V. The two flip endpoint variables.
+satisfies G < -b + 2/k for 0 < ell <= 1 and G >= -b with |G|/k bounded by
+2 beta^2 (ell^{-2} + ell^{-1}) for ell >= 1. Comparing G with D_0 gives the
+two stochastic inequalities on bounded intervals and tails.
-u_k = exp( -4 pi b / 9 * (k + theta / pi)^2 ). Eigenvalues
-of I_0.
+Evaluate the pure statistic on the Meixner--Pollaczek law. Normalize
+|Gamma(ell + i u)|^2 du to a probability on R; its Fourier transform is
+cosh(t/2)^{-2 ell}. The generating function (1 - i z)^{-ell + i u} (1 + i z)^{-
+ell - i u} = sum_n P_n(u) z^n gives real polynomials P_n of degree n with
+leading coefficient 2^n/n!, parity n, and squared norms c_n = (2 ell)_n/n!
+~ n^{2 ell - 1}/Gamma(2 ell). Evaluations:
-u_c = e^{-pi b / 4}. Central rate of 6.6.
+ (-1)^j P_{2j}(0) ~ j^{ell - 1}/Gamma(ell),
+ (-1)^j P_{2j}(i beta) ~ 2^{2 beta - 1} j^{ell + beta - 1}/
+ Gamma(ell + beta).
-w_e. The endpoint coordinate (marked bond site).
+The one-point intensity at zero and the pinned statistic give
-W_R. Arch mass at distance at least R in the half-plane.
-Equals R^{-1/4 + o(1)}.
+ rho_{Gamma, L}(0) ~ 1 + log L,
+ E_{Gamma, L}^pin prod_j (1 + k_j^{-2}) ~ L^beta / (1 + log L).
-x. See critical weight.
+For the unpinned orbital law, the two-sided pin bound and the Gamma
+comparisons give
-z = e^{-2 i phi}. The transfer-matrix eigenbasis variable.
+ c <= rho(0) <= C (1 + log M),
-Z(R). Planar nesting partition around the central face
-center in D_R, at fugacity 2. Equals R^{1/12 + o(1)}.
+with constants depending on the comparison parameters. For the pinned law,
+the product is decreasing, so the first Gamma comparison gives a lower bound
+of the pure statistic; conditioning on E and discarding later factors at
+least one gives the opposite, and P(E) = 1 - O(e^{-c m}) closes the loop.
+
+Order of limits. Fix b_+ > 6 alpha for the upper, b_- < 6 alpha and epsilon >
+0 for the lower, choose their ell parameters, and only then let M -> infinity.
+Put a_M = log F'_v(1) / log M. The pinning and residue bounds give
+
+ (1 - epsilon) b_-/pi <= liminf a_M <= limsup a_M <= b_+/pi.
+
+Letting b_-, b_+ -> 6 alpha and epsilon -> 0 gives
+
+ F'_v(1) = M^{3/4 + o(1)}. (**)
+
+Deficit and lower exponent. Define the positive 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)}.
+
+The coarse bound c_alpha sum_{d > 0} H(d) <= 1 from the convex balance, the
+inequality H_M(d) <= H(d), and delta_M -> 0 give saturation sum_{d > 0} H(d) =
+1/c_alpha. With A(n) = sum_{d >= n} H(d), this yields A(M) <= delta_M. Every
+arch contained within distance strictly less than M/2 of its source is
+disjoint from its nontrivial translates by M, so arches counted in the deficit
+reach distance at least M/2; hence delta_M <= W_{M/2}. Choosing M just above
+2 R gives W_R >= R^{-1/4+o(1)} and A(n) <= n^{-1/4+o(1)}.
+
+Wedge comparison. Intersect the upper half-plane with the wedge between rays
+of angles 0 and theta = pi/3. A source on its horizontal ray has horizontal
+distance a in {1/2, 3/2, ...} from the vertex. The signed balance holds on
+the infinite wedge: exhaust the half-plane by convex tessellation polygons
+with extending horizontal bases, use the convex-balance positivity for same-
+side arches, and pass to the limit using the absolute convergence of the fixed-
+ray exit sums.
+
+Put s = 3/8. For same-side walks the phase is e^{i s pi} to the left and
+e^{-i s pi} to the right, and exits to the other ray have phase e^{i s theta}.
+Let L_<(a) and L_>(a) be the lost left-side and right-side same-side arch
+weights on restriction from the half-plane to the wedge, including endpoints
+removed by the restriction. Subtracting the balances and projecting onto
+e^{i s theta} gives
-Z_R^chord = sum_{gamma in C_R} x^{N(gamma)}. Chord
-normalizer. Equals R^{3/4 + o(1)}.
+ sin(s(pi - theta)) L_<(a) - sin(s(pi + theta)) L_>(a) = 0,
-Z_{m,T}. The exact gas at period T and site count m.
+so L_<(a) = r L_>(a) with r = sqrt 2. For positive half-integer ports a, b,
+let D(a, b) be the lost same-side weight with those fixed endpoints. Reversal
+gives D(a, b) = D(b, a). The missing negative ports to the left contribute
+A(ceiling a), so
-zeta. A peripheral eigenvalue of E_0 (or E').
+ L_<(a) = A(ceiling a) + sum_{0 < b < a} D(a, b),
+ L_>(a) = sum_{b > a} D(a, b).
-End of PART 9. End of document.
+Summing L_< = r L_> over 0 < a <= T gives
+
+ (r - 1) sum_{0 < a < b <= T} D(a, b) + r sum_{0 < a <= T < b} D(a, b)
+ = sum_{0 < a <= T} A(ceiling a)
+ <= T^{3/4 + o(1)}.
+
+Since r > 1, this bounds sum_{0 < a <= T} (L_<(a) + L_>(a)) by the same order.
+After translating the source to a fixed port, the wedges increase with a, so
+their total loss decreases and
+
+ L_<(a) + L_>(a) <= a^{-1/4 + o(1)}.
+
+Two reflected wedges whose vertices are horizontal distance a on opposite
+sides of the source intersect in a triangle of diameter O(a). An arch reaching
+distance C a for a sufficiently large fixed C must leave at least one wedge,
+so its weight is at most the sum of the two losses, giving W_R <=
+R^{-1/4+o(1)}. This completes the boundary arch mass proposition. QED
+
+Extremal segments. Lemma. Fix a honeycomb vertex and two distinct height
+levels, with the fixed vertex on one level. The total critical weight of SAW
+segments from that vertex to any vertex on the other level, whose endpoints
+are opposite height extrema of the segment, is bounded by an absolute constant
+independent of the levels.
+
+Proof. Extend each endpoint to a port on the adjacent horizontal line beyond
+its extremal level. This requires at most one additional triangle at each
+end, strictly beyond the segment's range of heights: between two successive
+horizontal lines the lower triangles have a port on the bottom and the higher
+triangles have a port on the top; a triangle of either type has a neighbor of
+the other type. Choose an extension by a fixed local rule. Trimming recovers
+the segment, and possible choices have bounded multiplicity. The extended walk
+is a first-exit crossing of the slab between the two chosen lines. The convex
+boundary balance, exhausted horizontally, bounds its total weight for each of
+the finitely many possible starting ports. Removing the at most two extra
+vertices costs at most x^{-2}. QED
+
+For later use, split an n-edge walk at a highest vertex and treat the two
+halves. Within either half, go to the last occurrence of its lowest remaining
+level, then to the last occurrence of its highest remaining level, and continue
+alternately. These are segments of the lemma. Their positive integer spans
+decrease strictly after a possible equality of the first two, since each
+chosen extremum is the last occurrence of its level. The spans sum to at most
+2 n, so there are O(sqrt n) segments and at most (2 n + 1)^{O(sqrt n)} span
+lists. For a prescribed list, successive use of the lemma bounds the endpoint-
+summed product of segment weights by a constant to the number of segments,
+and recovering the weight of the joined walk costs x^{-1} per shared endpoint.
+These subexponential concatenation bounds will be used below in the marked
+argument, and they also recover the reciprocal exponential growth rate of
+full-plane SAW by summing over highest-vertex splits.
+
+Chords on the boundary. Let K be a finite convex tessellation polygon D
+sharing a starting port with the same inward direction as a containing convex
+cap D_0. Let L be the raw weight of first-exit chords of D_0 other than those
+confined to K and ending at a boundary port shared by K and D_0, and let E be
+the raw weight of first-exit chords of K ending on ports newly created by the
+restriction. Then
+
+ sin(alpha) L <= E <= L / sin(alpha).
+
+Proof. The common chords, confined to K and ending on a shared true boundary
+port, have the same intrinsic turning phase in both balances and cancel. The
+two remaining complex sums are equal. Every real coefficient in either convex
+domain lies between sin(alpha) and 1. Taking real parts proves both
+inequalities. The case with D_0 the saturated half-plane and K a finite convex
+cap follows by the same convex exhaustion used in the wedge argument. QED
+
+Visit identity. Let D be a finite convex tessellation polygon and v a
+honeycomb vertex not adjacent to a boundary port. Sum over all ordered first-
+exit boundary chords of D that visit v and set
+
+ S_D(v) = sum_{gamma ni v} x^{N(gamma)} cos(3 T(gamma)/8).
+
+Let Z_v be the partition of disjoint unoriented cycles in D strictly enclosing
+v and avoiding v, with weight 2 x^{|lambda|} per cycle. Let Z_out(lambda) count
+disjoint cycles in D surrounding and disjoint from lambda with the same
+weights. At a vertex of lambda at v, call the corner convex or reflex when its
+interior angle is 2 pi/3 or 4 pi/3, and put
+
+ c_lambda = sqrt 2 when v is a convex corner of lambda,
+ = 2 + sqrt 2 when v is a reflex corner.
+
+Proposition. The identity
+
+ S_D(v) = 3 Z_v + sum_{lambda ni v} c_lambda x^{|lambda|} Z_out(lambda)
+
+holds. Moreover, letting Z_D(f) denote the analogous partition around a face
+center f for each of the three incident face centers,
+
+ sum_{f ~ v} Z_D(f) = 3 Z_v + sum_{lambda ni v} 2 k_lambda x^{|lambda|}
+ Z_out(lambda),
+
+with k_lambda = 1 at a convex corner and 2 at a reflex corner, and therefore
+
+ (1/sqrt 2) sum_{f ~ v} Z_D(f) <= S_D(v) <= sum_{f ~ v} Z_D(f),
+ S_D(v) <= sum_{gamma ni v} x^{N(gamma)} <= S_D(v) / sin(alpha).
+
+Proof. Remove the triangle centered at v, creating three inner ports. For
+each outer starting port the signed balance still holds in the punctured
+domain: its source is outside every possible first-collision loop, including
+a loop surrounding the removed triangle, so the same collision cancellation
+applies. Subtracting the punctured balance from the original one cancels
+outer chords avoiding v, leaving the outer chords through v equal to the paths
+ending at an inner port. Take real parts, sum over the outer sources, and
+reverse the paths; reversal changes T to -T and preserves the real part. Thus
+S_D(v) is the real sum of paths from the three inner ports to the outer ports.
+
+For each inner source attach a background counted by Z_v and grow the weighted
+path tree in the punctured domain. Stop at an exit, a first self-collision, or
+first contact with a background cycle. The initial total strength of the three
+trees is 3 Z_v. A self-collision loop not enclosing v is approached from
+outside and cancels with its reverse exactly as in the convex balance. A loop
+enclosing v is approached from inside, since the inner source is in the same
+component as v. Its corner at the collision vertex is reflex: the other loop
+vertices contribute +-7 pi/3 of turning, and departure from the incoming stem
+contributes +-pi/3. The two orientations therefore contribute
+
+ 2 x^{|lambda|} cos((3/8)(8 pi/3)) = -2 x^{|lambda|},
+
+which cancels first contact with the same loop added to the background. The
+inverse operation, removing the first contacted (necessarily innermost)
+background cycle and following it from the contact in either orientation, is
+defined by first-contact without earlier stem intersection; every background-
+contact term is canceled exactly once.
+
+An outer exit can be reached with no background cycle at present, since the
+source starts inside every such cycle; its real sum is the S_D(v) identified
+above. An exit to another inner port, closed through v, gives a simple cycle
+lambda through v, and every remaining background cycle surrounds it. If
+L = |lambda|, opening the cycle at v leaves two directed inner-return paths
+of weight x^{L-1} each. At a convex corner their turns are +-5 pi/3, at a
+reflex corner +-7 pi/3: the full loop turn is +-2 pi with the turn at v
+removed. Write t_lambda = 5 pi/3 or 7 pi/3. The remaining real tree identity
+is
+
+ 3 Z_v = S_D(v) + sum_{lambda ni v} 2 x^{|lambda| - 1}
+ cos((3/8) t_lambda) Z_out(lambda),
+
+and moving the inner-return terms to the left supplies
+
+ -(2/x) cos(5 pi/8) = sqrt 2,
+ -(2/x) cos(7 pi/8) = 2 + sqrt 2,
+
+which is the first displayed identity.
+
+A cycle avoiding v places all three incident face centers in the same
+component as v; a cycle through v encloses exactly one incident face center
+at a convex corner and two at a reflex corner, and in a surrounding
+configuration must be innermost (an inner cycle avoiding v would also enclose
+v, lying on the outer cycle). This partitions the Z_D(f) configurations into
+the three cases and proves the second displayed identity. Comparing coefficients
+gives the (1/sqrt 2) two-sided visit comparison, and convex boundary positivity
+gives the ordinary visit mass bound. QED
+
+Flat-hexagon conclusions. Let Z(R) be the +2 nesting partition around the
+central face in D_R and, for honeycomb v in D_R, define V_R(v) as the raw
+chord mass through v. Then E_R^chord N = Z_R^chord^{-1} sum_v V_R(v). For
+v at distance at most R/10 from the center, every first-exit boundary chord
+through v is macroscopic for large R, so translating a hexagon of side
+floor(R/2) centered at each incident face inside D_R gives
+
+ V_R(v) >= c Z(floor(R/2)),
+
+uniformly for these bulk vertices. For v not adjacent to a boundary port,
+a translated hexagon of side 3 R contains D_R, giving V_R(v) <= C Z(3 R)
+uniformly. There are order R^2 bulk vertices and order R boundary-adjacent
+vertices, each of which contributes at most one. Hence
+
+ c R^2 Z(floor(R/2)) / Z_R^chord <= E_R^chord N
+ <= C R^2 Z(3 R) / Z_R^chord + C R.
+
+The chord partition Z_R^chord and the +2 nesting partition Z(R) will be
+identified later: Z_R^chord = R^{3/4 + o(1)} and Z(R) = R^{1/12 + o(1)}.
+Combining gives the mean E_R^chord N = R^{4/3 + o(1)} and bulk visit
+probability R^{-2/3 + o(1)}. The proofs of the two identities occupy the
+bulk of the paper.