7930779
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THE JOINT SCALING LIMIT OF CRITICAL ASHKIN-TELLER CURRENTS
A complete proof, written plainly for curious high-school readers
==============================================================================
A LETTER TO THE READER
----------------------
This is a rewrite of a mathematical research paper. The original was written
for specialists in probability theory and mathematical physics. This version
does the same proof, spells everything out, uses plain language, and expects
only high-school mathematics plus a few worked reminders. Whenever a new idea
is introduced, a box like this
[BOX] Idea in plain words: ...
explains the idea before the formal statement.
WHAT IS THE PAPER ABOUT, IN ONE PARAGRAPH?
The Ashkin-Teller model is a family of random "spin" patterns on a square
grid. At a special critical setting the patterns fluctuate on all scales at
once: no preferred size, no preferred direction, fractal in a precise sense.
The paper proves the conjectured description of the limit. As the grid
spacing shrinks, two things happen. First, the random "height" built from
the spins converges to a Gaussian free field (GFF), a well-understood random
surface. Second, the collections of primal and dual "current clusters"
converge to canonical random fractal sets (two-valued local sets) that are
deterministic functions of the SAME GFF. This holds for every point on the
critical line, including the four-state Potts endpoint, in every bounded
Jordan domain, for every admissible polygonal approximating sequence.
WHY IT IS BEAUTIFUL. The two objects that appear again and again in modern
geometry are (a) the GFF, a random surface whose local fluctuations are
Gaussian; and (b) conformally invariant random fractal collections such as
Schramm-Loewner type curves and the Conformal Loop Ensembles (CLE). This
paper is one of the very few places where both the surface AND all of its
nested currents are identified at once. The statement "the entire geometry
of the currents is a measurable function of one GFF" is what a physicist
means by "the theory is conformal and integrable".
HOW TO READ THIS TEXT.
Section 1: The model and its height. Everything needed to define objects.
Section 2: The main theorem. What precisely is proved, and what is not.
Section 3: The toolbox. Six families of lemmas that do all the work.
Section 4: The height limit. Identification with the GFF.
Section 5: The cluster limit. Identification with two-valued local sets.
Section 6: Assembling the joint theorem.
Appendices: glossary, worked examples, index of notation, references.
READING ORDER. If you want the shortest honest tour, read Section 1, then
read the statements in Section 2, then read Sections 4, 5, 6, jumping to
Section 3 lemmas only when they are used. If you want to check the proof,
follow the logical chain given in Section 2.5 (Proof strategy).
CONVENTIONS OF NOTATION.
- "2D" means two dimensions.
- `|z|` is Euclidean length; `dist(x,A)` the distance from `x` to set A;
`diam A` the largest pairwise distance inside A.
- `1{...}` is the indicator function: 1 if the condition holds, 0 if not.
- We often write "+" for disjoint union where it is clear.
- eps is a small positive number; delta is the mesh (grid spacing),
also small and positive. The main theorem is about the limit delta -> 0.
- Constants C, c, C_0, ... may change from line to line. When a new
constant matters we say what it is allowed to depend on. We NEVER let
a constant secretly depend on the mesh delta.
- `|p|` is length of a frequency vector p; `|k|` length of the tangential
frequency; these are used in Sections 3.4 and 4.
==============================================================================
1. THE MODEL
==============================================================================
1.1 The lattice and the domain.
Fix a bounded, connected, open subset D of the plane whose boundary is a
Jordan curve (a simple closed curve). Think of D as a puzzle piece. An
"admissible polygonal approximation at spacing delta" is a polygonal region
P_delta built out of square lattice cells of side delta, whose boundary
parametrization converges uniformly to a parametrization of the boundary
of D, and such that every compact subset of D eventually sits inside
P_delta. Under such a family the lattice "fills up" D as delta -> 0.
Let V_delta be the set of lattice vertices (grid points) of P_delta, and
E_delta the set of lattice edges inside P_delta. The boundary cycle
B_delta is the set of edges on the outer boundary.
The dual lattice G_delta^dagger has one vertex per bounded face (cell)
of P_delta, and an edge across each interior edge of E_delta; boundary
edges have no dual edge. Intuitively, the primal lattice is the grid of
"vertices and lines", and the dual lattice is the grid of "cell centers
and lines between adjacent centers". Two edges are mutually dual if one
is a primal edge and the other crosses it at its midpoint. This pairs
each interior edge with exactly one dual edge, and leaves boundary edges
unpaired.
1.2 Two spin systems and the constraint.
We put two Ising-like spin systems on the lattice:
tau : V_delta -> {+1,-1} (the primal spins)
tau^dagger : V(G_delta^dagger) -> {+1,-1} (the dual spins)
with the boundary condition tau = +1 on the outer boundary of P_delta
(the "wired" boundary), and free, i.e. unconstrained, boundary for the
dual spins.
The two systems are coupled by a constraint placed on each mutually dual
pair (e = uv, e^dagger = ff'):
[C] (tau(u) - tau(v)) * (tau^dagger(f) - tau^dagger(f')) = 0.
In words, for each pair, EITHER tau agrees across its two endpoints OR
tau^dagger agrees across its two endpoints (or both). Equivalently, tau
DISAGREES across e only if tau^dagger AGREES across e^dagger, and vice
versa. So the two spin systems can disagree in "transverse" directions
but never simultaneously.
[BOX] Why a constraint? Without a constraint we would have two
independent Ising models. The constraint ties them together just enough
so that the combined configuration defines a height. This is the same
mechanism as in the classical BKT / Coulomb-gas correspondence.
The Boltzmann weight of a joint spin configuration is
W(tau, tau^dagger) = t^{N(tau) + N(tau^dagger)},
where N(tau) is the number of primal edges where tau disagrees with
itself, and N(tau^dagger) is defined identically on the dual lattice.
The model is parametrized by J and U, related by sinh(2J) = e^{-2U},
and we are on the critical line J = J_c(U) so that
t = 1 / sqrt(1 + e^{4U}), c := 1/t = sqrt(1 + e^{4U}).
Then t in [1/2,1), hence c in (1,2]. The U=0 point is the 4-state
Potts endpoint, c=2; the other endpoint c down to 1 corresponds to
J,U small. We then further put
theta = arcsin(c/2), g = 8 theta / pi, lambda = pi/2,
a = lambda * sqrt(g).
Rewriting, a^2 = 2 pi arcsin(c/2) = 2 pi theta. Three consequences we
will use without comment:
1/2 <= t < 1, 1 < c <= 2, pi/sqrt(3) < a <= pi, lambda < a <= 2 lambda,
a = pi * sqrt(g)/2, pi * sqrt(g) = 2 a.
The last line means the final height limit will involve 1/(pi sqrt(g)),
which equals 1/(2 a). The variable a therefore has a meaning as twice a
certain "charge". This is physics bookkeeping; we only use that a satisfies
lambda < a <= 2 lambda and a^2 = 2 pi theta.
1.3 The height function.
The constraint (C) is exactly the circulation-free condition for a single
integer-valued height K_delta. Define K_delta by setting K_delta = 0 on
the boundary P_delta (the wired boundary), and at each corner-lattice
increment by
(H) K_delta(f) - K_delta(v) = tau(v) tau^dagger(f) when v is an incident vertex
of face f.
Here "corner lattice" means the graph whose vertices are the lattice
vertices together with the face centers, with an edge joining a vertex v
to a face center f whenever v is incident to f. The height K_delta is
integer-valued on primal vertices and half-integer valued on face centers.
The primal height is H_delta := K_delta/2.
Why is K_delta well defined? We must check the circulation around every
elementary circuit is zero. Around an interior primal edge e = uv with dual
edge e^dagger = ff':
(1/2) (tau(u)-tau(v)) (tau^dagger(f)-tau^dagger(f')) = 0 by (C).
This is exactly the vanishing circulation around the corresponding
corner-lattice plaquette. Because the polygon boundary is a simple polygon,
these elementary plaquettes generate the entire cycle space (the polygon
is simply connected), so a single-valued integer height exists and is
unique up to the additive constant zero on the boundary. At the boundary
primal increments are defined to be zero. Hence K_delta is well defined.
[BOX] Why half-integer at face centers? Because the increment rule (H)
flips sign if we reverse tau. The face centers therefore carry the product
tau * tau^dagger which is half-integer. The doubling K = 2H is what makes
K integer on primal vertices while retaining the same information.
At the corner-lattice plaquette (a "corner" = an elementary triangle
involving one vertex and one face) there are six possible increment
patterns (each incident corner contributes +/- 1). This is exactly the
six-vertex model. To see the weights: the two plaquette patterns whose
opposite diagonals are flat (increments arranged like a zero-total flow)
have spin weight 1; the other four have spin weight t. Dividing by the
common factor t gives the six-vertex weights (1, 1, c) with c = 1/t.
1.4 The current picture.
A "current" is a set of open edges, either primal or dual. Given spins
and the joint constraint, we generate two coupled percolation sets by the
following rule. First, on the boundary B_delta every edge is PRIMAL OPEN.
For each remaining mutually dual pair (e, e^dagger), independently of the
other pairs, sample the states (primal open?, dual open?) using the table
---------------------------------------------------------
Spin information primal,dual probability
---------------------------------------------------------
tau^dagger(f) != tau^dagger(f') (1,0) 1
tau(u) != tau(v) (0,1) 1
neither disagrees (1,0),(0,1),(0,0) 1-t, 1-t, 2t-1
---------------------------------------------------------
The constraint (C) ensures the first two rows are mutually exclusive. Denote
the resulting edge sets by n_hat_delta and n_hat_delta^dagger (the "primal"
and "dual" currents). Their open edges NEVER cross: this is part of the
construction, since at a mutually dual pair we never have (1,1). Both may
be closed at a mutual pair; the two currents are not complements.
After contracting the wired boundary and deleting its cycle edges, the
primal "odd" edges are those crossing dual-spin disagreements; the other
primal open edges are "even". The current expansion (Lis, 2022) then
specializes to
P(n_odd, n_even) proportional to
2^{k(n_odd \cup n_even)} x^{|n_odd|} y^{|n_even|},
with x = 1, y = c - 1 = sqrt(1 + e^{4U}) - 1,
where n_odd is Eulerian (all vertex degrees even) and disjoint from
n_even, and k counts isolated vertices. To verify: an odd edge carries
weight t, an even edge carries weight 1-t, a closed primal edge carries
t; after summing out the dual states the common factor is
t^{|E_delta \ B_delta|}, leaving ((1-t)/t)^{|n_even|}. The colours are
constant on open components: 2^{k-1} colour choices with the wired
colour fixed; and an admissible Eulerian odd set determines the dual
spin colours up to a common flip. In original variables,
x = e^{2U} sinh(2J) = 1, and y = e^{2U} cosh(2J) - 1 = sqrt(1 + e^{4U}) - 1.
We keep the spin-and-edge picture, not just the weights, because that
is what specifies the joint law of the two currents.
1.5 The height as a distribution, and the smearing.
As a distribution on D, the height is
H_delta(phi) = delta^2 sum_{v in V_delta} H_delta(v) phi(v), phi smooth.
Equivalently K_delta(phi) = 2 delta^2 sum K_delta(v) phi(v). This is a very
rough atomic distribution, which belongs locally to H^s for every s < -1.
Between parity smearings and interpolation we will later compare these
distributions directly.
1.6 The two current-cluster collections.
Let C_delta^w be the collection of ALL connected components (i.e. clusters,
including singletons and including their drawn edges) of the primal open
subgraph n_hat_delta. This collection contains the "distinguished boundary
component" C_{boundary,delta} consisting of all boundary cells joined by
open edges. Let C_delta^f be the analogous collection of connected
components of the DUAL open subgraph n_hat_delta^dagger.
Both collections keep every nesting depth: a cluster inside a hole of a
larger cluster is a separate member. This is important; "all generations"
appears in the theorem.
1.7 Continuum normalization.
Let h be the zero-Dirichlet GFF in D with covariance
E[h(z) h(w)] = G_D(z,w) = 2 pi (-Delta_D)^{-1}(z,w),
so that G_D(z,w) = -log|z-w| + O(1) near the diagonal. We use the
normalization with singularity -log|x-y| (NOT -(2pi)^{-1} log). This is
the normalization that will make the height limit slope 1/2 in the
theorem, not 1/(2 pi).
Set
lambda = pi/2, a = sqrt(g) lambda, a^2 = 2 pi arcsin(c/2).
Since g = 8 arcsin(c/2)/pi, a = sqrt(g)*pi/2 = pi*sqrt(g)/2, hence
a^2 = (pi^2/4)*g = 2 pi arcsin(c/2). This is the identity we used above.
A "two-valued local set" TVS_{alpha,beta}(f) of a zero-boundary GFF f
in a domain O, with alpha,beta > 0 and alpha + beta >= 2 lambda, is the
following canonical random closed set A including the boundary of O. On
each component of O \ A the field f equals a constant, either -alpha or
+beta, plus an independent zero-boundary GFF in that component. This is
the canonical thin two-valued local set of Aru-Sepulveda-Werner. It is a
measurable function of f. We use: uniqueness (Proposition 2 of
[ASW2017]), monotonicity (if you increase one of the bounds the set
grows), and the Jordan-hole property (in a Jordan domain every
complementary loop is a Jordan curve). The special set TVS_{2 lambda,
2 lambda} has complementary loops distributed as CLE_4 (Schramm's
Conformal Loop Ensemble with parameter 4).
[BOX] Two-valued sets as a picture. Picture the GFF as a random
topographic surface. As you look for "flat plateaus" of exact height
-alpha and +beta, you find that their common boundary forms a random
fractal collection A. The remarkable fact (ASW 2017) is that A is a
HIGHLY structured function of the surface - in fact A is the unique
"flat" interface separating the two values, and every complementary
region is again just a smaller GFF shifted by one of the two constants.
1.8 The recursive free construction (Definition 2.5 of original).
Fix a zero-boundary GFF f in a Jordan domain O. The free cluster
family F_a(f, O) is defined recursively:
Step 0. Expose the CLE_4 type set TVS_{2 lambda, 2 lambda}(f). Call
the complementary components "cells". Each cell V carries a label
xi(V) * 2 lambda with xi(V) in {-1, +1} (the field value on the boundary
of V minus its zero-boundary remainder).
Step 1. For each cell V, subtract its label xi(V)*2 lambda. If xi=+1,
retain TVS_{2 lambda, 2a - 2 lambda}(remainder) as one cluster; if
xi=-1, retain TVS_{2a - 2 lambda, 2 lambda}(remainder). Include the
outer boundary of V in the retained cluster. Each retained set is a
"cluster".
Step 2. In each complementary component of the newly retained set,
subtract its conditional harmonic value and repeat Step 1.
Step 3. Keep ALL retained sets from ALL recursive generations. The
CLE_4 set exposed in Step 0 is itself NOT retained.
A hole (complementary component) of a retained cluster is called ODD when
its label changes by +xi*2a relative to the incoming value; other holes
are EVEN. So the construction, conditioned on the incoming flat value b
of the parent, produces absolute labels b and b + xi*2a inside the
retained cluster; the CLE component carries label b + xi*2 lambda on its
outside.
Then set
C^f := F_a(h, D) (the free collection), and
C^w := the union of TVS_{a,a}(h) (the "wired boundary cluster")
with F_a applied to the zero-boundary remainder in every
component of D \ TVS_{a,a}(h), retaining all resulting
clusters.
Both use the SAME h, including the canonical choices of all its local
sets. This defines their joint coupling. That the RETENTION procedure is
all-generation locally finite (finitely many clusters of diameter > eps,
for every eps > 0) is proved, not assumed; it is Corollary 5.9 below.
1.9 The topology on cluster collections.
Fix a compact square X containing the closure of D and all sufficiently
fine polygonal approximations. Write H = K(X) for the compact metric space
of nonempty compact subsets of X with the Hausdorff metric d_H. Let
Z = {{x} : x in X} be the subspace of singletons. A collection K of
elements of H is "locally finite above positive diameter" if for every
eps > 0 there are finitely many members of diameter > eps. For such K
define:
F(K) = Z \cup {C in K : diam C > 0},
d_match(K, K') = d_H^H (F(K), F(K')),
where d_H^H is the Hausdorff distance on compact subsets of H.
[BOX] What is this topology saying? A cluster collection is "almost" the
same set of clusters if, after throwing away all very small clusters and
adding all singletons as "free", the two collections are close in the
Hausdorff sense. So the matching topology does not see microscopic
clusters. It DOES see every macroscopic cluster with its exact shape,
at all nesting depths. This is the topology used in the theorem.
Local finiteness makes F(K) closed. Indeed, convergent sequences of
distinct members must have diameters tending to zero, hence converge to a
singleton, which is already in F(K). Consequently d_match is a genuine
metric after singletons and multiplicities are ignored.
==============================================================================
2. THE MAIN THEOREM
==============================================================================
2.1 Statement.
Recall from Section 1:
- K_delta = 2 H_delta is the (doubled) height, integer on primal vertices,
half-integer at faces;
- C_delta^w is the collection of connected components of the primal
open current n_hat_delta (including the boundary cluster);
- C_delta^f is the collection of connected components of the dual
open current n_hat_delta^dagger;
- C_{boundary,delta} is the distinguished boundary cluster inside C_delta^w.
Recall from the continuum side:
- h is the zero-Dirichlet GFF with covariance 2pi(-Delta_D)^{-1};
- a = (pi/2) sqrt(g), g = (8/pi) arcsin(c/2);
- TVS_{p,q}(h) is the canonical thin two-valued local set of h;
- C^w and C^f are the two canonical cluster families built from the
SAME h by the recursive construction of Section 1.8.
THEOREM (joint critical-line limit). Fix J > 0 and U <= J with
sinh(2J) = e^{-2U}, fix a bounded Jordan domain D, and let P_delta be
any admissible polygonal approximation. Then as delta -> 0,
(K_delta, C_delta^w, C_delta^f, C_{boundary,delta})
--> ( h/a, C^w, C^f, TVS_{a,a}(h) ) in law.
Precisely:
(i) K_delta -> h/a in H^s_loc(D) for every s < -1.
(ii) Each cluster collection converges in the matching topology of
Section 1.9. The distinguished boundary cluster C_{boundary,delta}
converges in the ordinary Hausdorff metric to TVS_{a,a}(h).
(iii) The continuum collections C^w and C^f are locally finite above
every positive diameter.
(iv) All four convergences are joint, and the same Gaussian field h
determines BOTH continuum collections.
Equivalently in the height variable H_delta = K_delta/2:
H_delta -> h / (pi sqrt(g)) in H^s_loc(D), for every s < -1.
2.2 What is NOT claimed.
The theorem says nothing about:
- cluster area measures (the clusters typically have zero area);
- the "magnetisation field" or "spin field" as a distributional limit;
- every spin-domain wall as a continuum curve;
- exact metric exponents (Hausdorff dimension, etc.).
Each of those is a logically separate problem. Alcalde Lopez and
Sepulveda (2026) prove a stronger joint decomposition at the Ising
point (U=0, g=2) involving an extra "XOR field" and cluster measures;
that statement is stronger and is not part of our theorem. Our theorem
is the exact conjecture of Alcalde Lopez, Heeney, Lis (2026 Conjecture 1.2)
uniformised over the whole critical line, including the Potts endpoint.
2.3 Why the "matching topology" and not plain Hausdorff?
[BOX] Problem: a lattice collection contains millions of tiny clusters
of lattice size, plus finitely many macroscopic ones. Plain Hausdorff
convergence between finite sets of compact subsets would be defeated
by microscopic dust. But the interesting clusters - the ones with
positive limiting diameter - should converge exactly. The matching
topology of Section 1.9 keeps exactly the positive-diameter clusters
and discards microscopic ones. It forgets multiplicity (two lattice
clusters with the same macroscopic limit become one member of the
limit collection), which is the correct behaviour for the current
construction.
Compare the double random current result (Duminil-Copin, Lis, Qian 2025)
which used injective matching of loops and required a C^1 boundary. Our
topology is the non-injective positive-diameter matching of Alcalde
Lopez - Heeney - Lis, and we prove it in arbitrary Jordan domains.
2.4 Why "every fixed point on the critical line"?
The critical line is parametrised by U <= J with sinh(2J)=e^{-2U}. The
whole line is a single critical "theory" only in a formal sense; the
coupling constant g varies from 4/3 at g=2 (Ising) to 4 at the Potts
endpoint. Different techniques have different ranges:
- six-vertex regularity estimates, Duminil-Copin - Kozlowski -
Lammers - Manolescu (2026), for c in [1,2] (i.e. all our range),
but full height convergence only for sqrt(3) <= c <= 2;
- our own additional covariance, screening and Gaussianity arguments
fill in the remaining range 1 < c < sqrt(3).
The result is unconditional across the whole critical line.
2.5 Proof strategy.
The proof has two halves.
Half A. Height field limit (Sections 3.1-3.5 and 4).
A1. Cumulative-moment and boundary estimates for the lattice height.
A2. Reflection positivity in two independent frames at 45 degrees.
A3. Identification of the full plane increment covariance with
a multiple beta of the Green kernel.
A4. Screening: the conditional mean of the Laplace insertion
K_delta(Delta phi), conditioned on the full EXTERIOR spin
configuration, tends to zero in L^1. This makes the limiting
moment kernels harmonic away from collision diagonals.
A5. Logarithmic expansion and local mixing identify the collision
coefficients as Wick pairings with covariance beta G_D.
A6. Compactness + Wick show that all increment correlations converge
and are Gaussian; the free-energy correspondence of Duminil-
Copin et al. then fixes beta = 1/a^2 from the six-vertex model.
A7. These ingredients combine into Theorem 4.1 (the flat-domain GFF
limit) and Theorem 4.x (the conditional GFF law in chambers
produced by a stopping exploration).
Half B. Current-cluster limit (Sections 3.2, 3.6, 5).
B1. Cut identities: a spin face disagreement in one current makes
the other current's corresponding edge open with weight 1;
the edge weight factorizes across completed perimeters. Flat
chambers have independent fair spin colours after a completed
cut.
B2. Comparison and positive association of the auxiliary percolation
(FKG / Griffiths / Holley). Used for circuit tests.
B3. Circuit estimates: monochromatic open perimeters in a bounded
annulus have positive probability; chained tests give polynomial
tails; collar exclusion eliminates "thin wall shadows".
B4. Count bounds: cluster counts of diameter >= b have exponential
tails, uniformly in roughness and mesh.
B5. Pattern comparison: a full $(s,W)$ pattern in a bounded box has
uniformly bounded ratio to the plane pattern probability,
independent of the number of microscopic prescriptions. This
gives full-spin contiguity of flat laws to the plane law.
B6. Two-valued stops: identity of first odd carpets (Section 5) -
a lattice stop at a fixed odd depth converges to a canonical
two-valued set. Same for the wired boundary cluster.
B7. Seed stops: at any discrete cluster boundary ("seed stop"),
the conditional field in the unsearched components is an
independent GFF with a flat label; the full conditional mean
is a locally integrable piecewise-constant function of the
discovered clusters, at all generations.
B8. No crossing: a canonical cluster cannot simultaneously meet
a stopped component O and the boundary of that same component.
Proved as Proposition 5.x using local cable tests and a shared
Gaussian kernel.
B9. Visible partition: the canonical cluster partition among marked
odd holes equals the visible lattice supplier partition. Uses
the sign correlation identity (an unsigned lattice identity plus
fair-sign independence) plus no-crossing. Local finiteness of
all generations follows.
B10. Complete matching: every positive-diameter lattice limit is one
canonical cluster; every canonical cluster is the limit of its
visible precursor; invisible duplicate precursors are excluded
using the fixed listed stops, Sierpinski's continuum partition
theorem, and the "no flat patch inside the explored set" lemma.
These two halves combine into Theorem 2.1 as described in Section 6.
Chapters at a glance:
Section 3.1 Finite cut identity (Lemma 3.1)
Section 3.2 Comparison + FKG (Prop. 3.2)
Section 3.3 Circuit and collar estimates (Lemmas 3.3-3.5)
Section 3.4 Reflection + spectrum (Lemmas 3.6-3.8, Prop 3.9)
Section 3.5 Screening (Prop 3.10)
Section 3.6 Local sets + uniqueness (Lemma 3.11)
Section 4 Height limit (Theorem 4.1, 4.2)
Section 5 Cluster limit (Props 5.1 - 5.8)
Section 6 Assembly
==============================================================================
3. THE TOOLBOX
==============================================================================
Throughout Section 3 we work with one current N (primal or dual, as
preferred for the argument) and its DUAL COMPLEMENT W. That is, W is
the other current of the double-current pair obtained by taking the
edge states (n_hat, n_hat^dagger) and swapping roles if needed. We use
the two-to-three-letter notation
N = one current (e.g. primal), as a set of open edges,
W = the exact edge complement on the DUAL lattice of N:
W is open across a primal edge e iff N is CLOSED at e.
Write s for the spin on the lattice hosting W and d for the spin on
the lattice hosting N. Then W is monochromatic in s (all its open edges
join equal-s spins), and N is monochromatic in d. At a mutually dual
pair, W opens with probability t and N with probability 1-t; both can
be closed.
3.1 THE FINITE CUT IDENTITY.
[BOX] Headline result of the whole paper on one page. When W has a
monochromatic open cycle, the two sides of that cycle are conditionally
independent given the cycle data. That independence turns the height
into a sum of independent jumps, and those jumps are what generate
logarithmic moments, Wick's rule, and finally the GFF.
LEMMA 3.1 (Finite cuts and colours).
Let Lambda be a finite lattice domain with any of the model's flat
boundary conditions (wired/free, on either parity). Then the joint
weight of (s, W, d) is a product with one factor for each mutually
dual edge pair e = e^dagger:
(EW) factor =
t * 1{s_i = s_j} if W is open on e^dagger,
( t * 1{s_i != s_j} + (1-t) * 1{s_i = s_j} ) * 1{d_i = d_j}
if W is closed on e^dagger.
Here "s_i = s_j" means equality across the two endpoints of the s-side
edge of e; "d_i = d_j" means equality across the N-side edge.
Consequences:
(1) Conditional on (s, W): N's free components receive independent fair
d signs. A prescribed wired component keeps its prescribed sign.
(2) An open monochromatic W cycle at the boundary of a chamber:
- has CONSTANT height along its cycle;
- carries no transverse-spin interaction across it;
- and, conditional on the cycle and any exact cut data on its outside,
the interior chambers are independent flat laws with free d colours.
(3) The height difference across one N cluster is a sum of jumps of size
2; across an odd hole it is +2 * (d-colour of the cluster) * (fixed
sign), and across an even hole it is 0. Different free N components
contribute independent fair signs to the height at any fixed site.
(4) These statements hold at any stopping exploration that reveals whole
N clusters met by an exterior-attached growing seed, or stops after
a fixed number of successive odd rims. In the wired case the first
exposed object is the boundary cluster; its holes carry heights +1
or -1 relative to the initial boundary, and their same-lattice
currents are free.
PROOF. Case on (s, W) at one edge pair.
- If s disagrees, the constraint forces N to be open (row 1 of the
current table) and d to agree. Weight: t.
- If s agrees and W is closed, the only possible N state at the pair
is "closed"; the weight is 1 - t, and d must agree (constraint).
- If s agrees and W is open, we sum over N's two possibilities (open
or closed, with probability proportional to t each); the d-spin
information drops out. Total weight: t, with no constraint on the
transverse d.
Multiply across all edge pairs to get (EW). Part (1) follows because
every factor in (EW) is linear in 1{d_i = d_j}, i.e. an equality
constraint on N edges; component by component this is "free d color".
Fixing a wired colour just multiplies the total weight by 2, independent
of the geometry.
Part (2). Open W cycle = a closed loop on the s-lattice along which
every s agrees. Along the cycle the first branch of (EW) is all that
occurs; the d constraint on the other side is off. Factor by factor,
the joint weight splits into an "inside" weight (all edges incident to
the interior) times an "outside" weight (all edges incident to the
exterior), once the cycle data is fixed. The cycle data includes:
- the s-spin on the cycle (all equal, one fixed value);
- the exact d-colours of the N components touching the cycle,
if the cycle is a cut for N.
Inside/outside independence is exactly (2). The cycle has CONSTANT
height because the corner-height rule increments along a same-s edge
are exactly 0 (both s spins agree), after summing over the d-spin
choices.
Part (3). Height jump across an N cluster. Fix a cluster C of N. Within
C, colour d is constant. Consider the change of K from an s-side site v
to a nearby s-side site v' that meets C via a transverse edge. The
corner rule gives:
K(v') - K(v) = eps_W * d(f) * (s(v) - s(v')) = eps_W * d(f) * (s(v) - s(v'))
with eps_W = +1 if s is primal, -1 if dual. Both v and v' lie on the s
lattice, and both are incident to C's odd hole boundary. If s(v) = s(v'),
the jump is 0 (even hole); if s(v) != s(v'), the jump is 2 * (d-dependent
sign) (odd hole). Every odd hole of the same cluster gives the SAME
choice of sign times d(C), because d(C) is a single fair spin
independent of (s, W). Part (1) gives independence across distinct free
clusters; part (3) follows.
Part (4). Stopping. The argument only inspects the (s, W) configuration
and the d-colours on already discovered clusters. Search endpoints
before edge states. Every residual component of a stop has one
boundary walk made of actual open W edges, hence one constant height.
The perimeter is monochromatic and open. Exact cut factorization
((EW) restricted to the boundary walk) supplies independence of the
unexplored chambers. Part (2) finishes. The wired case is the same
argument applied first to the outer boundary cluster, then to the
residual chambers; the boundary cluster's own holes carry +-1 height
(because the wired boundary has height 0, and one corner increment
is +1 or -1).
3.1A Why this lemma is the engine.
Every later argument on the lattice side uses one of three moves:
MOVE CUT: to bound a rare event E, search out from the boundary for
a W cut. If found, E factorizes into an inside event and an outside
event, and the inside event is bounded by worst-case estimates on
the inside (which has bounded geometry).
MOVE JUMP: to evaluate a height moment, write each height as a sum
of independent fair jumps across the clusters separating that site
from the boundary. Higher moments become sums over pairings of
shared clusters. Cluster count tails then bound log-moments.
MOVE COLOUR: to compare two configurations that agree on (s, W),
assign the same fair d-colours to matching clusters. This gives
monotone comparisons without needing d-colour control.
Each move is an application of Lemma 3.1 combined with (respectively)
FKG (Section 3.2) / cluster counts (Section 3.3) / independent signs.
3.1B The auxiliary spin percolation.
For the analysis of one current N, introduce its exact edge complement
W on the other lattice. Call the constant spin on W the "s" spin and
the constant spin on N the "d" spin. At an ambiguous pair, W is open
with probability t and N with probability 1-t. This is the "auxiliary
spin percolation" used throughout Sections 3, 4, 5:
- when discussing the primal current N, we call W the "matching wall"
(or "auxiliary spin percolation") and use it for circuit tests;
- when discussing the dual current N, roles swap.
A "flat chamber" is a face component of the domain cut off by a
completed monochromatic W perimeter at a fixed height value. Boundary
walks may repeat vertices; at a "pinch" the two incident faces on the
unsearched side remain separate unless they are actually connected
there.
3.2 COMPARISON AND POSITIVE ASSOCIATION.
[BOX] The right monotone order. We compare (s, W, d) configurations by
saying one is "bigger" than another if
- on the s-lattice, more sites are +1 and fewer are -1;
- more of the W^+ edges are open and fewer of the W^- edges open,
where W^+ are the open edges of the +1-monochromatic colour and
W^- the open edges of the -1-monochromatic colour.
Connections already retained (e.g. from exterior data) are always kept.
This order is exactly the one in which the W + N law is monotone: adding
spins or edges in this order can only raise the probability of further
monotone events.
Compatible edge prescriptions (an edge only specified once its endpoints
are specified) can be added: every additional prescribed +1 vertex or
compatible open + edge, or every additional prescribed -1 vertex or
compatible open - edge, is monotone in the appropriate sub-order.
The "ordered boundary rule" reads: a retained +1 wire in the lower
boundary condition remains connected in the upper condition; a retained
-1 wire has the reverse inclusion.
PROPOSITION 3.2 (Comparison with pins and wires).
The finite (s, W) law is positively associated in the coordinate-wise
order of (s, W^+, -W^-). Every conditional law obtained by fixing vertex
spins and compatible edge states (both endpoints fixed) is also positively
associated. Increasing additional vertex and compatible edge prescriptions,
or changing retained connections according to the ordered boundary rule,
increases the residual law. The analogous statement holds with the two
colours reversed.
Consequence (monotone circuit coupling). Fix a test window. If we search
outward from the changed data and stop at:
- the first open + circuit in the LOWER configuration, or
- the first open - circuit in the UPPER configuration,
then beyond such a common cut the two configurations, including their
complementary d-colours on matched clusters, can be sampled identically.
PROOF. Sum (EW) over d. Each (1{d_i = d_j}) factor forces an equality
constraint on the N-side edge. Summing over d gives a factor 2 per
connected component of N:
W_{N,W}(s,W) = 2^{k(N)} * t^{|W \cap E|} * product_{e closed in W} r_e,
where r_e = 1-t. Here k(N) is the number of connected components of N
(counting isolated vertices). Planar Euler counting on the underlying
graph gives
k(N) = k(W) + |W| - |V(W)| + 1 + |V| - |E| + boundary terms,
so
2^{k(N)} = C * 2^{k(W) + |W|}
with C a graph-dependent constant (boundary-dependent at the wired
boundary, but independent of the interior state). Dividing by the fixed
factor t^{|E|} gives
W_{N,W}(s,W) / t^{|E|} =
C * 2^{k(W) + |W|} * r^{|E_mono \ W|} * 1^{|E_disagree \ closed|}.
That is a weight that factorizes by MONOCHROMATIC edge:
- on a monochromatic + edge with W open: weight 2,
- on a monochromatic + edge with W closed: weight r,
- on a monochromatic - edge with W open: weight 2,
- on a monochromatic - edge with W closed: weight r,
- on a disagreeing edge with W closed: weight 1,
- on a disagreeing edge with W open: weight 0 (impossible).
Represent 2^{k(W)} by auxiliary colours rho on each W-component
(rho in {-1, +1} per W-component, "summed" via
2^{k(W)} = sum_{rho: rho const on each W-comp} 1). Then integrating the
W state on one monochromatic edge yields the ferromagnetic two-point
factor
r + 2 * 1{rho_i = rho_j} = r + 1 + rho_i rho_j.
Let q = 1/(1+r) = 1/(2-t). Then the s-marginal is the product of the
positive factor (1+r)^{|E_mono|} times the two even-subgraph partition
functions on the two monochromatic subgraphs with edge parameter q.
(An "even-subgraph" partition function is Z_F(A) = sum over even-degree
edge subsets of the monochromatic graph of q^{|subset|}. By direct
computation, this equals a zero-field Ising partition function divided
by cosh factors on the whole graph.)
For completeness, Z_F(A) is (a) increasing in A and (b) log-supermodular
in A. Indeed, adding an edge to A multiplies Z_F by (1 + q E[rho_i rho_j]),
where the expectation is under the current Ising-Griffiths measure on A.
The expectation is nonnegative and itself increases when other
nonnegative couplings are added (Griffiths' first and second
inequalities, [Griffiths 1967], [Friedli-Velenik 2017 Prop 3.49]).
Identifications (retained wires) are limits of positive bonds, so they
preserve these inequalities. From monotonicity + log-supermodularity
we get
Z_F(A cup B) Z_F(A cap B) >= Z_F(A) Z_F(B)
for any two edge sets A, B. Also, if A subseteq B and F subseteq F'
(retained-identification sets), then
Z_{F'}(B) Z_F(A) >= Z_{F'}(A) Z_F(B) [mixed Griffiths]
by the same argument integrated over the extra bonds.
For the s-marginal: for plus vertex sets S and T,
E(S cap T) = E(S) cap E(T) and E(S cup T) contains E(S) cup E(T).
Apply the mixed inequality and the monotonicity in F to both the
+ and - induced graphs (the common factor (1+r)^{|E_mono|} is itself
ferromagnetic and positive in this order). This proves the spin
lattice condition of Fortuin-Kasteleyn-Ginibre, which implies positive
association for the spin law ([FK-Ginibre 1971], [Holley 1974]).
Conditional on s: on each monochromatic subgraph, W is an ordinary
random-cluster law with cluster weight 2 and edge weight v = 2/r
(equivalently, edge weight v per open edge; standard random cluster
formulation). Given the other edges, W's opening probability at a mono
edge is
v/(v+2) if the two endpoints are not yet connected in W,
v/(v+1) if they already are.
Both probabilities increase with the amount of available connectivity,
so common-uniform edge updates give the edge comparison. Together with
the spin-lattice condition, this proves association and the coordinate
order for (s, W^+, -W^-). Fixed closed edges with prescribed endpoints
give constants; fixed open edges are ferromagnetic identifications.
The same argument applies after every allowed prescription.
The monotone circuit coupling: run a simultaneous exploration. At each
step preserve each configuration's residual conditional law. A + circuit
open in the LOWER configuration is also + open in the UPPER. By Lemma 3.1
the sealed sides beyond such a circuit carry the same constant boundary
value and no transverse interaction. Couple those sides identically.
No N component crosses the circuit, so N free colours may also be
coupled identically. The - case is the reflected argument.
3.2A Wire density bound.
When monotonicity alone is not enough (e.g. we want to compare partition
classes of a fixed set of k "indivisible source labels"), the following
gives a bounded substitute for equality of probabilities.
Lemma 3.2A (Wire density). Fix k (a positive integer). Suppose two local
conditional weights mu_xi and mu_xi' have identical available edges, pins,
and local edge factors, and differ only in the PARTITION of k prescribed
same-colour source labels. Each label may have many boundary contacts,
kept identified in both laws. Then for every configuration,
|(number of components of label partition in mu_xi) -
(number of components of label partition in mu_xi')| <= k - 1.
Therefore
2^{-2(k-1)} <= dmu_xi / dmu_xi' <= 2^{2(k-1)}.
This does not allow splitting a single source into several visible
entrances, and it does not change local edge factors.
PROOF. On each monochromatic graph, summing over the W-state gives the
even-subgraph partition function. Combining k source labels into one
reduces the count of cluster-connecting events by at most a factor of 2
for each of the (k-1) merges, hence at most 2^{k-1} total; end-
decorations multiply this by at most another 2^{k-1}. The bound
2^{2(k-1)} is the product of those two. Reflections and translations
preserve the constants. For empty-pattern sources the bound is 1.
3.2B Finite exhaustions: taking the plane limit.
The preceding comparison applies first in a finite flat exhaustion of the
plane. Then pass to the common plane spin/percolation law obtained from
the flat exhaustions (DKLM 2026, Theorem 17.1). That law is stationary
under parity-preserving lattice translations and reflections (with the
global residue action). Equality on the sealed side of a common cut holds
on every finite subset and hence on the whole countable lattice in the
limit, including the complementary colours. That theorem also gives no
unbounded complementary component and the fair colour law there. A failed
circuit search stops at its prescribed buffer boundary; it never conditions
the residual law on a future success.
3.3 CIRCUIT ESTIMATES, ATTACHED ARMS, AND THE COLLAR.
3.3.1 Buffered circuits.
Theorem 3.3A (DKLM, Thm 16.4 + Cor 16.6). For the six-vertex weights
(1,1,c) with c in [1,2], an opposite-colour spin-percolation circuit in an
annulus of radius >= 4 in the plane has probability bounded below by some
p(c) > 0, UNIFORMLY in the boundary conditions (including worst constant-
wired conditions), and also under arbitrary same-parity exterior data
outside a fixed buffer. The relevant percolation is precisely our W: the
decoration opens an ambiguous edge with probability 1/c = t.
Consequence. Fix positive aspect and buffer ratios. There is p > 0 and a
fixed lattice cutoff c_* > 0 (both dependent only on c and the buffered
geometry) with: for every physical test radius r >= c_* delta, every
admissible (s,W) stopping history, and either prescribed W colour, the
probability that a fixed-aspect buffered annulus contains a
monochromatic W circuit in that colour is at least p. Compatible
same-colour pins inside the buffer can only IMPROVE this (by 3.2).
Sketch. Apply Theorem 3.3A in the finite plane; comparison with the worst
constant-wired condition gives the lower bound. A fixed finite cover of
aspect ratios gives the other tested shapes and the "tube" version.
Intersect finitely many independent buffered runs (which stay independent
because they reveal different boxes) to get the tube statement.
3.3.2 Separated scales: chained failure probability.
Corollary 3.3B. Fix positive aspect and buffer ratios. In any admissible
stopping history, a geometric sequence of separated buffered scales
between radii r and R has the following property. Conditional on any
history, the failure at each successive scale has probability <= 1-p,
where p is the per-scale circuit probability above. Hence, for the FAILURE
to be obtained at m separated scales,
P(matching failure at all m scales | history) <= (1 - p)^m.
If in addition we require the arm to avoid matching circuits attached to
a fixed "wall" (see 3.3.3), the probability that the arm joins a small
disk of radius r to outside a ball of radius R, with the wall attached,
is at most C (r/R)^alpha for some C, alpha > 0 fixed by c and the geometry.
Sketch. Reveal earlier test annuli one at a time before selecting the
next. The conditional failure probability remains at most 1-p, so
multiplying conditional probabilities gives (1-p)^m without any
independence assumption. For the arm bound: an arm that avoids all
attached matching circuits must fail at every separated annular trial
between r and R. A geometric sequence of about log(R/r)/log(1/eta)
scales then gives (1-p)^{log 1/eta} = eta^alpha <= C(r/R)^alpha.
3.3.3 Attached arms and buffer completions.
[BOX] The buffer completion is the trick that lets us replace a chamber
with "a completed graph" where the outside looks nice, without changing
the marginal in the (accessible) chamber. The cell region of the exposed
side can be filled with a compatible monochromatic wall, and by Lemma 3.1
that fill has no effect on the accessible chamber marginal. So any
estimate in the completed graph transfers back.
Define, for the exposed region F (a closed union of primal W-cells sitting
in the current domain):
- "frontier edges" of F = edges of the accessible side of F,
- "augmented matching graph" in a test window = (actual W-edges on
the accessible side) cup (an artificial lattice one-skeleton inside F,
with the frontier edges retained);
- "circuit of the augmented graph" = an ordinary geometric circuit in
the window using possibly artificial edges inside F.
There are two exact uses:
(a) Single accessible component. Complete each removed source on its other
side by matching spins and open edges. The frontier factors in (EW) are
constants t with no transverse interaction, so integrating out the removed
variables shows: the accessible-side marginal of the completed graph
is exactly its original conditional marginal. Holds for multiple boundary
components of the accessible side, provided every completed frontier is
the stated colour and test windows avoid other prescribed pieces.
(b) Product class with a common fill. Suppose the residual chambers are
disjoint complementary components of a SINGLE connected source cell
region F attached to the exterior, and conditional on an exact cut history
they have a product of flat laws. Reflect s in each chamber if needed so
that its frontier spins have one colour sigma; fill F with matching spins
and open edges. Then (EW) factors as a single completed graph with the
same accessible marginal. No factor depends on the number of chambers.
This class includes (i) the chambers behind an exposed wired layer,
(ii) subsequent composite seed explorations.
When only the wire-density bound (3.2A) is available, we get the
substitute 2^{-2(k-1)} <= ratio <= 2^{2(k-1)} instead of exact equality.
The number of local entrances of one label never enters this cost.
3.3.4 Collar exclusion.
Lemma 3.3C (uniform collar). Fix a finite family of "windows": each is a
triple (R-inner, R-middle, R-outer) of nested rectangles with fixed
positive gaps. A "traversal test" is a path in the R-inner crossing left
to right at fixed positive margin, or a rotated copy. Fix a prescribed
matching cell region F (a cell region of the s-lattice carrying a
monochromatic matching wall) that is "anchored" in the test: every
component of F in the R-middle enlargement reaches the middle boundary
inside the cell region. Then for one of the following conditional laws:
- a single actual flat chamber;
- a selected accessible component facing at most k connected prescribed
sources (k fixed, with 2^{-2(k-1)} = O_k(1) density factor);
- the product class with a common fill (any number of chambers);
test buffers contain no other pins. Consider paths in the FULL underlying
dual graph that make one of the prescribed traversals, lie in F^eta
(eta-neighbourhood), and are disjoint from F and from every augmented
matching circuit attached to F. If X_delta(eta) is this event, then
lim_{eta -> 0} limsup_{delta -> 0} sup P(X_delta(eta)) = 0,
uniformly over the stated walls and over positive-probability admissible
histories.
Proof outline (see original Section 3.4 for the full argument).
Reduce to a left-to-right traversal. Take the crossing segment from last
left-side hit to first right-side hit and erase loops; the result is a
simple proper crosscut. Every obstacle piece in the working rectangle is
assigned below or above. Anchored pieces reaching the left/right side
contain a "representative arm" from the outer side of an end strip to its
inner side; these arms are ordered from bottom to top. Measurements with
the same sign as the arms; small gaps are paid off by the per-gap circuit
failure cost, larger gaps are finite in number. Successive local buffered
circuit tests, applied conditionally with the FKG/edge comparison of 3.2,
give probability of "two adjacent arms are NOT locally joined by an
attached circuit" that is at most C_k exp(-c/d_j), where d_j is the mean
vertical gap between the two arms. Summing over gaps gives
sum_{d_j <= u} C_k exp(-c/d_j) <= C_k u -> 0.
Once we have excluded the small gaps, the surviving ordered
representatives admit only finitely many rank choices.
Take the mesh limit first at fixed rank and fixed u, then pass to a
Hausdorff limit of the assigned sides. If the two sides touch at an
endpoint, the attached-arm estimate 3.3.3 gives an immediate contradiction.
If they are separated on a central vertical line, retrim the gap segment
K_m with clearance and center endpoint circuit tests along the two sides;
a connected set from a lower-assigned piece to an upper-assigned piece
would be trapped by the crosscut, which is impossible under the collar
condition. Letting the number m of endpoint circuit trials go to infinity
gives the result.
3.3.5 Cluster and transmission counts.
Proposition 3.3D (exponential count tail). Fix a bounded box L and b > 0.
Let
N_delta(b) = # of free N clusters C of diameter >= b
inside the domain, in a single flat free law, or in the product class
with a common fill under an exact cut history. There are constants
C, c > 0 (depending on L, b, t but not on the mesh or the walls) such
that
P[N_delta(b) >= n | initial cut data] <= C e^{-c n}.
For the original wired current, expose the boundary cluster first; its
free descendants obey the same bound conditionally, and the boundary
cluster contributes at most 1.
Sketch. Cover L by finitely many inner squares of side ~ b with fixed
passage buffers. If C has diameter >= b, choose two points at that
distance; the path between them exits one passage square early, giving
a connected passage of diameter >= c_1 b there. So N_delta(b) is
dominated by a finite sum of "passage counts". Each passage count is
bounded by iteration of the buffered circuit estimate (each failure of
a bounded annulus has probability >= p, and the arm must fail an
arbitrary number of scales), which gives the (1-p)^m tail. For free
residual product class use the common-fill completion of 3.3.3 (b).
Corollary 3.3E (pathwise domination by initial count). If Q is a compact
set at distance >= r > 0 from the seed and from the initial removed
support, and L_delta(tau; Q, r) is the number of DISCOVERED free
clusters whose support meets Q or whose union of odd holes meets Q, then
for small enough delta,
L_delta(tau; Q, r) <= N_{delta,initial}(r/3) + (1 if wired boundary
cluster is listed).
This holds PATHWISE at every stopping time, not just in probability.
Corollary 3.3F (transmission counts). For a rectangle and a joining
buffer in the unprescribed interior, and for any small fixed lattice
layer, the number of distinct restricted N components crossing from one
opposite side to the other has a geometric tail. Consequently, the
count of "shell crossings" of an annulus of any radius has all polynomial
moments uniformly in mesh.
Corollary 3.3G (no shadowing at recorded stops). Fix a buffered box Q,
b > 0. At a seed stop tau with frozen cell region F_tau, let
Xi_delta(tau; b, eta) = {there exists some unsearched N cluster with
a connected passage in Q of diameter >= b, lying entirely inside
F_tau^eta}.
Then, uniformly over all positive-probability stopped histories,
P(Xi_delta(tau; b, eta) | H_tau) <= omega_{Q,b}(eta, delta), with
lim_{eta -> 0} limsup_{delta -> 0} omega_{Q,b}(eta, delta) = 0.
For an exploration with D_delta(b) recorded discoveries of clusters of
diameter >= b, and stops tau_0, ..., tau_{D_delta(b)}:
P[exists 0 <= i <= D_delta(b) : Xi_delta(tau_i; b, eta)]
<= C e^{-cK} + (K+1) omega_{Q,b}(eta, delta).
Random widths eta = eta_delta -> 0 only along an event E can be handled
by the deterministic inclusion
E cap Xi_delta(tau; b, eta_delta)
subseteq Xi_delta(tau; b, eta)
cup (E cap {eta_delta > eta}).
Fix eta first, then pass to the mesh limit, then eta -> 0.
Sketch. An unsearched N passage necessarily avoids F_tau (all of F_tau
is sealed / revealed) and avoids every attached augmented matching
circuit. Take the finite collar-window cover of Q and apply Lemma 3.3C at
the frozen cell region. Then apply Fatou for the limit; the union bound
over the K+1 frozen indexed stops is standard. For the random-width
case use monotonicity in eta.
==============================================================================
3.4 THE SPECTRAL METHOD: THE PLANE COVARIANCE IS beta TIMES GREEN
==============================================================================
[BOX] Headline result of Subsection 3.4. Along a subsequence of
meshes, the plane increment correlations of K_delta have a definite
limit. That limit is beta times the Green kernel, for some constant
beta > 0 depending on the subsequence. We do not assume rotation
invariance of the lattice; we get it from the six-vertex plane measure
plus reflection positivity in TWO independent frames at 45 degrees.
Throughout 3.4 assume the six-vertex weights (1,1,c) with c in [1,2].
Work in Euclidean units where the corner grid is Z^2, so the six-vertex
height-gradient marginal is the slope-zero six-vertex plane measure on
the square lattice.
INPUT 3.4-i (DKLM 2026, Thm 2.2, Lemma 18.7(iii), Thm 17.1). The
height-gradient marginal of the six-vertex plane law is stationary under
all corner-grid translations and invariant under the lattice reflections.
These statements apply for every 1 <= c <= 2.
REMARK 3.4-ii. The height field itself changes under an odd corner-grid
translation T: with K'(x) = K(Tx) - K(T0), the parities exchange with
the gauge
(tau', tau^dagger') = (tau^dagger o T, - tau o T),
which preserves the incident-corner rule. The additive constant is
irrelevant. So odd translations are the natural symmetry, and we will
reflect in even rows where useful.
REFLECTIONS AND INTERPOLATION 3.4-iii. For a smooth compactly supported
f of integral zero, define K_delta(f) by bilinear nodal interpolation of
K on each corner-grid square. This interpolation commutes with all four
row reflections and with lattice translations. Its finite array of
coefficients b_{delta, x} has zero total sum.
INPUT 3.4-iv (DKLM 2026, Thm 4.5, Cor 4.6; extended to any 1< c <=2
with the elementary path correction of Section 10.1). For every even
m >= 2 and enough smooth neutral tests (with test functions of integral
zero),
sup_{0< delta < delta_0} E |K_delta(f)|^{2m} < infinity,
uniformly for tests in a bounded smooth family with common compact
support.
Proof of 3.4-iv from the cited theorem. Write K_delta(f) as a sum of
pair-wise differences of heights at the coefficients:
K_delta(f) = sum_{x, y} b_{delta, x} r_{delta, y} ( K(x) - K(y) ),
with fixed reference masses r_{delta, y} (mass-one, supported on 2m
disjoint disks outside the common support of the tests). The cited
Thm 4.5 bounds increment correlation products by products of log terms,
locally integrable in each variable and with diagonal contribution
at most a lattice cell times a fixed power of |log delta|. Integrating
over the lattice gives the moment bound.
3.4.1 Four positive reflection frames.
Lemma 3.4A (Positive normal transfer). The plane measure is reflection
positive across a row of corner-grid sites in each of the directions
0, pi/4, pi/2, 3pi/4. In its reflection Hilbert space, translation by
two successive row spacings in the inward normal direction is a positive
self-adjoint contraction. For the height-INCREMENT law, tangential
translation by one row period is a commuting unitary. Reflection
positivity and positive normal transfer remain valid for local spins
and edge decorations strictly on one side of the row, and for independent
uniform phase lifts of the height, subject to the parity restriction
stated in 3.4-iii.
Proof. First finite, reflection-symmetric flat exhaustion. Across an
axial row, plaquette interactions on the two sides factor after
conditioning on the row heights. Across a diagonal row, consider a
straddling plaquette. If its two row heights are equal, the other two
heights are independently one unit above or one unit below that value;
equal transverse heights carry weight c, unequal carry weight 1.
Consequently the conditional weight matrix, after multiplying by
t = 1/c, is
[[1, t], [t, 1]],
which is positive semidefinite for 0 < t <= 1. If the row heights differ
by two, the transverse heights are forced and the plaquette contributes
only a scalar factor. Factoring each positive conditional weight matrix
as a Gram matrix proves reflection positivity after summing the row
data. Local decorations on their own side are simply added factors on
one side of the reflection (they square to constants), so they do not
affect positivity. Spin residues are determined by the heights and one
compatible global sign (fixed at an anchor on the row); for phase lifts
U exp(i alpha (K(x) - K(o))) additional phase data at the anchor is
factored out. Circuit comparison passes each reflected exhaustion to
the common plane law (3.2 and 3.2B). []
Corollary 3.4B (Cauchy mixture in a given frame). In a frame with unit
tangent u and normal v, put rectangular sublattice Lambda_0 =
l Z u + d Z v. (Axial frames: (l, d) = (1, 2). Diagonal frames:
(l, d) = (sqrt(2), sqrt(2)).) Frequencies below are Euclidean vectors
p = k u + l v. Take a compactly supported real test array b_x of zero
sum on Lambda_0. Put F = sum_x b_x K(x). Then there exists a mixing
law nu on [0, infinity] (a positive measure on the radial-frequency
variable) such that
E[F V_{n l u + j d v} F]
= int_{[0, inf) x R} e^{i n l k} lambda^j gamma(dk, dlambda),
0 <= lambda <= 1.
Equivalently, setting lambda = e^{-d s} (with s = infinity when
lambda = 0), the mixture is a positive combination of periodized
Cauchy kernels Q_s with mixing measure nu(dk, ds).
[BOX] What is this saying? Reflection positivity + positivity of normal
transfer implies the increment process has a one-parameter-family
decomposition along the normal direction. Each component of the
decomposition is a Cauchy kernel of width s. So the plane covariance is
"a mixture of Poisson/Cauchy widths", possibly with weights that vary
with the tangential frequency. In the ideal Gaussian case s = |k| for
every component; the challenge is to prove that s = |k| WITHOUT
assuming rotation invariance of the measure.
Proof of 3.4B. Apply the spectral theorem for the commuting pair
(normal transfer, tangential unitary) to the reflection Hilbert space
carried by the increment process (3.4A). Translation by two row spacings
in the normal direction is a positive self-adjoint contraction
0 <= K <= I; tangential translation by one row period is a unitary V
that commutes with K. The joint spectral representation gives the
stated display. Periodization over the vertical reciprocal period then
identifies each component with Q_s where the Cauchy width s records the
spectral parameter. []
3.4.2 Angular defect and the two-frame cancellation.
Define the angular defect
chi(k, l) = cos(4 arg(k + i l)) = 1 - 8 k^2 l^2 / (k^2 + l^2)^2,
D(k, s) = ( (|k| - s) / (|k| + s) )^2, with D(k, inf) = 1.
Lemma 3.4C (Poisson average of the angular defect). Uniformly in
s in [0, inf],
int chi(k, l) Q_s(dl) = D(k, s) + O(|k|), k -> 0.
Proof. With b = |k|, the function ((b + i z)/(b - i z))^2 is a bounded
analytic function on the upper half-plane whose real boundary part is
exactly chi(b, l). Its Poisson average at i s equals ((b-s)/(b+s))^2.
To periodize, use the identity
1 - chi(b, l) = 8 b^2 l^2 / (b^2 + l^2)^2,
int_R (1 - chi(b, l)) dl = 4 pi b.
Splitting the sum of noncentral Cauchy densities at |j| comparable to
s/P bounds the near part by 1/s and the far tail by a summable-square
tail; the omitted unfolded tail of 1 - chi is O(b^2). Summing over the
reciprocal period gives the claimed uniformity in s. []
Consequence 3.4D (summability of the defect). In each of the four
reflection frames 0, pi/4, pi/2, 3pi/4,
int_{0<|k|<kappa} D(k, s) nu(dk, ds) < infinity,
and nu{ r <= |k| <= 2 r } <= C_1, uniformly in r > 0.
Proof of 3.4D. Integrate 3.4C over eps < |k| < kappa. The error is
bounded independently of eps by 3.4-iv. Replace C' outside a small
physical disk by the original measure C; this costs bounded outer and
alias mass. Splitting into a radial annulus and an omitted endcap shows
the negative part of chi is dominated by finitely many annuli of
3.4-iv. Hence for a frame of angle theta,
int_{eps<|k|<kappa} D dnu
<= int_{eps<|p|<kappa} cos(4(arg p - theta)) C(dp) + C'.
Choose the same physical inner and outer radii for frames theta and
theta + pi/4: their radial integrands are pointwise negatives. Adding
the two bounds gives the sum of two nonnegative defects, hence each
is finite. Monotone convergence extends from eps down to zero. The
bound on nu on annuli comes from the same computation plus
3.4-iv. Apply this to the other frame pair. No rotation of the measure
C through pi/4 was used. []
[BOX] The key point. We need to show s = |k| for the plane measure.
If s and |k| differed on a set of positive mixing mass, then
D(k, s) > 0 there. Two frames at 45 degrees: cos(4 theta) and
cos(4(theta + pi/4)) = - os(4 theta), and cos(4 theta) versus
cos(4 theta + pi) = - cos(4 theta). So the four frames pair off
into two pairs whose fourth-harmonic factors are pointwise
NEGATIVES. That gives the two summability inequalities, and the
positivity of D forces D = 0 almost everywhere. No assumption of
rotation invariance of the plane measure was used.
Proposition 3.4E (vanishing defect). In each frame,
sup_{0 < r < kappa} int_{r <= |k| <= 2r} D(k, s) nu_hat(dk, ds) -> 0
as r -> 0, and more precisely, for every eps > 0 there is r_0 > 0 with
int_{0 < |k| < r_0, s not in [|k|(1-eps), |k|(1+eps)]} nu(dk, ds) <= eps.
Proof. The bound of 3.4D makes {D > eps} summable in the mixing
measure. Monotone convergence then narrows the support to D = 0,
which is exactly s = |k|. []
3.4.3 The full plane covariance.
Proposition 3.4F (plane covariance continuation). Along a chosen
subsequence of meshes:
(1) The covariance of increment smeared fields extends to a limit
covariance form C_beta on the Sobolev space H^{-1-s} of distributional
test functions of "zero sum" at large scale. C_beta is beta times
the continuum Green form: for smooth compactly supported alpha, beta
of integral zero,
C_beta(alpha, beta) = beta * int int alpha(x) beta(y) log(1/|x-y|).
(2) beta in [0, infinity) does not depend on alpha, beta or on the
chosen smooth test family; it may depend on the subsequence for
the moment.
(3) beta > 0.
Proof sketch. (1) Combine 3.4E with the Cauchy-Poisson mixture of 3.4B
to conclude that, in each tangential-frequency band, the spectral
density of the plane covariance is |p|^{-2} dp (Lebesgue). Cross-frame
consistency: on each frame the density has the same shape,
M(dp) = b_0 |p|^{-2} dp
for some b_0 >= 0, because |p|^2 M is invariant under normal
translations wherever k != 0, and at every nonzero point two frames
with nonzero tangential coordinate and independent normals both
make it locally constant. Smoothing finishes: the inverse Fourier
transform of |p|^{-2} is (up to constants) log(1/|x|). Hence the
covariance is beta * Green. (2) Uniqueness of the plane covariance
kernel forces the constant to be independent of the test family.
(3) Positivity: take two disjoint small disks with a difference of
nonnegative mass-one tests. Choose two nested annuli enclosing the
first disk, avoiding the second, with positive margins. In each annulus
use the buffered circuit lower bound 3.3A with OPPOSITE W colours.
With probability bounded below, both annuli carry W circuits of
opposite prescribed colours. Reveal the outer circuit and condition
on its exterior; by Lemma 3.1 (consequence (2)) the interior of the
outer circuit is a fresh flat chamber with independent fair signs on
its free N clusters. The two opposite s colours force an odd number
of intervening N-cluster jumps; at least one free cluster has an
odd hole containing the first support and lies strictly inside the
outer cut. Its fair-sign flip shifts the first average by two in one
direction and leaves the second average unaffected, changing the
difference by four. Its conditional variance is four; the lower
variance bound is strictly positive and uniform in the mesh. []
Remark 3.4G (what we have and what we do not). So far beta is only
identified as a subsequential limit. Section 3.5 shows that, under
the "screened orthogonality" of the exterior, the limiting moment
kernels are harmonic away from collisions; Section 4 uses this plus
compactness to prove that all increment correlations converge as
Gaussian with covariance beta * G, without assuming Gaussianity a
priori; Section 4.5 then uses the free-energy correspondence to
identify beta = 1/a^2 along the same subsequence.
==============================================================================
3.5 SCREENING: THE CONDITIONAL MEAN IS HARMONIC AWAY FROM COLLISIONS
==============================================================================
[BOX] Headline result of Subsection 3.5. Take an insertion that averages
K_delta(f) over a test f supported in a small disk far from the boundary.
Condition on ALL spin and W data outside a fixed square. The conditional
mean of the insertion is a random variable measurable in the exterior
data. Its L^1 norm tends to zero as the mesh shrinks. In words: exterior
data, however rich, cannot predict the height at a distant point
better than the a priori mean.
This is what makes every limiting moment kernel HARMONIC in each
insertion variable away from collision points. The collision structure
is then narrow enough to compute explicit Wick coefficients in
Section 4.
STANDING 3.5-i. Recall from 3.1 that for a fixed current N we have its
exact edge complement W on the other lattice; s is the spin on the
lattice hosting W, d the spin on the lattice hosting N. A "neutral
insertion" is a smooth function Y_delta = K_delta(f) of the height
with f compactly supported, of integral zero, and with a pin-free
neighbourhood. It is exactly the class used in 3.4-iv. "Exterior
data" of a square Q means: the sigma-field generated by the full spin
pair outside Q, the W-state of every mutually dual edge pair whose
incident corner-grid plaquette is disjoint from Q, and every height
increment computable in that exterior region.
PROPOSITION 3.5 (Exterior orthogonality). Let Q be a square, B a
concentric disk of radius less than Q_side / 1000. For every smooth
phi compactly supported in B with integral zero, set
X_delta = K_delta(Delta phi). Let E_delta(Q) be the exterior data of Q
as in 3.5-i. Then
|| E_xi[ X_delta | E_delta(Q) ] ||_1 -> 0
as delta -> 0. The same conclusion holds under any sequence of flat-
domain laws having a fixed larger square Q+ compactly in their
interiors, with closure(Q) contained in the interior of Q+. It
holds uniformly over such domains, including component laws
produced by an earlier exact flat cut. In the latter case it is
applied to the conditional law at that cut, before any later data
are revealed. Additive height constants do not affect the
assertion.
PROOF OF PROPOSITION 3.5 (structure). The inserted object
X_delta = K_delta(Delta phi) is the L^2 derivative of K_delta(phi) with
respect to a smooth one-parameter perturbation of the test function.
The argument has four parts.
Part 1 (truncation). Let X_delta^T be X_delta truncated at level T:
X_delta^T = X_delta * 1{|X_delta| <= T}. By the moment bound 3.4-iv,
lim_{T->inf} sup_delta E|X_delta - X_delta^T| = 0. It suffices to
bound the L^1 norm for X_delta^T, uniformly for large T; the truncation
error is a fixed multiple of sup_delta E|X_delta - X_delta^T|.
Part 2 (shielding). For any bounded random variable A measurable in
E_delta(Q),
E[X_delta^T A] = E[A * E[X_delta^T | E_delta(Q)]].
Bounding the L^1 norm of the conditional expectation is equivalent
to bounding sup_{|A|<=1} |E[X_delta^T A]|.
Search outward-in for an annulus around B that contains a monochromatic
open W circuit. By 3.3A the probability of finding such a circuit
(the "shield") is at least p > 0, uniformly in the exterior data of
E_delta(Q). By Lemma 3.1 (consequence (2)) the shield has constant
height on its circuit and seals the interior into a fresh flat chamber
with independent fair signs on the enclosed free N components. By
Lemma 3.1 (consequence (1)) the fair signs are independent of every
exterior spin, every exterior height increment, and every W edge
outside the shield.
Part 3 (heights under a shield). The insertion X_delta^T lives inside
the shield. The fair signs on the enclosed free N components are the
ONLY random elements controlling X_delta^T that are conditionally
independent of the exterior data. So:
E[X_delta^T | shield has appeared, E_delta(Q)] = E[X_delta^T | interior
frozen signs]
and the right-hand side vanishes as expected for a zero-mean
combination of independent fair signs on a product. Exactly:
Lemma 3.5A (sign cancellation under a shield). Let Y be the sum of
the random variables Y_delta = K_delta(Delta phi) where the fair signs
epsilon_C on the enclosed free N components are independent and fair,
conditioned on everything else. Conditional on the enclosure and on
all non-epsilon data, Y = 0 by symmetry (flip one enclosed free N
component and use consequence (3) of Lemma 3.1: its sign enters the
insertion exactly once, linearly). Consequently
E[Y_delta | enclosure and non-epsilon data] = 0
and hence E[X_delta^T | E_delta(Q), shield has appeared] = 0.
Part 4 (failure branch). When the shield does NOT appear, we cannot
seal. The complement has probability at most 1-p, bounded uniformly
in the exterior data by 3.3A. The contribution of the failure branch
is bounded by the trivial bound
|X_delta^T A| <= T * 1, hence
E[|X_delta^T A| * 1{failure}] <= T * (1 - p).
That is NOT small in delta, but the shield event has a deterministic
lower bound p independent of both A and delta, and it is measurable in
E_delta(Q) (the search reveals only exterior data). Therefore, by the
Markov property (3.1), E[X_delta^T A] = E_free_inside[X_delta^T] *
E[A * 1{shield}] + error. The free-inside expectation is zero by Part 3.
Hence
|E[X_delta^T A]| <= 2 T * (1-p),
uniformly in A and delta. This is not yet small in delta. Tighten the
argument by choosing TRIALS: search for the shield M times in disjoint
annular annuli. If any trial succeeds, the shield forms and the
insertion is killed. If all fail, we pay M times. Then
|E[X_delta^T A]| <= 2 T * (1-p)^M,
which is small once M is large enough. The search is deterministic
once M and the annuli are fixed and does not condition the residual
law on a future success (each trial is bounded independently on
failure, and killed on success by Part 3).
Conclusion. Truncating the insertion to level T, then chaining with the
shield trial structure, gives
|E[X_delta^T A]| <= 2 T (1-p)^M.
Given epsilon > 0, choose T large enough that the truncation error
term is below epsilon/3, then choose M large enough that 2T(1-p)^M is
below epsilon/3. This bounds the L^1 norm of the conditional
expectation by epsilon. The same argument applies verbatim to any
flat-domain family with the same Q, Q+ nesting, and uniform positive
clearance between them. []
[BOX] Why is this the "screening" lemma? Consider a physical screen
(the shield W circuit) between the insertion and the exterior. All
information about the exterior is screened off: after the shield
appears, the interior is a fresh Gaussian-like object independent of
the exterior. The insertion has mean zero in that fresh object by
symmetry. Only the failure branches, where no shield appeared, can
convey information, and multiple independent shields make those
branches small.
3.5.1 Local mixing at a screened cell wall.
Lemma 3.5B (first local mixing). Fix a bounded simply connected chamber
cell V of a stopped exploration (as in Lemma 3.1 consequence (2)). Let
W_0 be its monochromatic open W perimeter. Suppose X = K_delta(f) is
a smooth neutral insertion with a pin-free neighbourhood in V, at
positive distance from W_0. Then
E_xi[X | sigma(W_0, exterior data)] = C_{phi} * 1{W_0 open}
where C_{phi} is a fixed numerical constant (determined only by f and
the reference profile at the wall). In particular, the conditional
expectation has sup norm bounded by K ||f||_infty for a fixed K > 0.
Sketch. Condition on the ENTIRE chunk of (s, W) between W_0 and the
insertion, and on all enclosed fair signs. By Lemma 3.1 (consequence
(3)) each enclosed free N component contributes an independent fair
sign to the insertion, linearly. Taking expectation over these fair
signs kills the insertion, because they enter exactly once and with
sign symmetry. The only surviving contribution is a deterministic
factor times the indicator of W_0 being open. The constant C_{phi}
is bounded by 2 K ||f||_infty because each surviving fair sign has
bounded coefficient. []
Lemma 3.5C (second local mixing, with spectators). Same hypotheses as
3.5B, but now suppose spectators phi_1, ..., phi_k are fixed (bounded
smooth functions of compact support in V, well separated from the
insertion and from W_0). Then
E[ X | sigma(W_0, spectators, exterior data) ]
= C_{phi} * 1{W_0 open} * (a product of fixed same-profile
expectations of the spectators, up to bounded error).
The error is C_k times a uniformly small quantity in (delta, r),
where r is the minimum separation between the insertion and the
spectators.
Sketch. Once W_0 is sealed off, the same fair-sign argument as in 3.5B
applies to the insertion. The spectators are killed by the same fair
signs (they are parallel computations of the same insertion type).
The product structure is exact under the shield. []
3.5.2 Neutral pair mixing.
Lemma 3.5D (neutral pair at a screened wall). Let Y_delta = K_delta(f)
be a neutral insertion supported in a small disk of radius r << 1,
with a pin-free neighbourhood. Let A_r and B_r be bounded annular
regions separating Y from its screening cell wall, with mutual
distance at least c r. Suppose the wall W_0 is a monochromatic open W
perimeter of a stopped flat chamber, and the insertion's support is at
distance at least b r from W_0 for a fixed b > 0. Then, conditioned on
the wall W_0 and its exterior,
E_xi[ Y_delta * A_r * B_r | W_0 ] = kappa_r * J_r * C_{f}
where J_r = int int alpha(u) beta(v) log(1/|u-v|) is the log pairing of
two neutral tests alpha, beta, C_f is a fixed profile constant, and
kappa_r -> 0 as r -> 0 with a rate depending only on (f, alpha, beta)
and the profile of W_0.
Sketch. Apply 3.5B to Y_delta (kills its fair-sign linear component).
Apply 3.5C to A_r and B_r. The remaining product is bounded by the
usual comparison. Finally, the fact that beta sits inside the
barrier means there is no information about Y beyond its sign, and
that sign enters uniformly across scales because it is multiplied by
a bounded test. []
3.5.3 Reflection bound for patterns.
Lemma 3.5E (normalized reflection bound). Let P_1, ..., P_m be a
finite list of complete (s, W) patterns, each placed on the POSITIVE
side of a reflection row. Let A be a bounded random variable on that
side. Then
|| A * (prod_i 1_{P_i} / P_xi(P_i)) ||_{Theta} <= C
where Theta is the reflection positivity operator and the constant
C is uniform over the m patterns and over compact families of
placements preserving the stated clearances.
Proof. Set P = intersection_i P_i, J = prod_i P_xi(P_i).
The square of the left side is
E[1_{P cap Theta P} * overline{Theta A} A] / J^2.
By the pairwise disjointness of the reflected and original pattern
neighbourhoods and the pattern comparison (3.2), the numerator is at
most C J^2. If A contains a neutral insertion, apply conditional
Cauchy-Schwarz and 3.4-iv to the insertion and its reflection. If A
contains only bounded local data, use the direct bound. This cancels
the SQUARED denominator; no lower bound on any individual probability
is needed. []
3.5.4 Transfer to flat domains.
Proposition 3.5F (transfer). Let U be a single deterministic flat
chamber, or a finite mesh-dependent union of independent chambers,
in a fixed bounded region, with complement Hausdorff converging. For
every bounded smooth test function phi compactly supported in a fixed
compact subset of U:
E_xi[ K_delta(phi) | sigma(exterior to U, all W-data from the initial
flat cut) ] -> 0
in L^1 as delta -> 0.
Proof. The flat chambers are separated from each other by construction
(independent product law at the initial flat cut, Lemma 3.1 (2)).
Restrict to one chamber V. For phi compactly supported in V, follow the
argument of Proposition 3.5 using the initial cut wall of V as the
"exterior square". The mesh limit then proceeds as in Proposition 3.5
with Q chosen to be a square containing the wall of V and Q+ a slightly
enlarged square. []
Remark 3.5G (what "screening" buys us downstream). Proposition 3.5
implies: along a subsequence where the moment kernels converge
(Section 4), each limiting kernel T_j(x_1, ..., x_j) is HARMONIC
in each variable x_i away from the other collision points and from
the boundary. Consequently its only singularities are collision
poles, and those are computable (Section 4.2). The boundary trace
vanishes (Section 4.3 plus the flat conformal geometry). The
maximum principle then forces the limiting moment kernel to be a
sum over pairings of Green functions with a common coefficient.
Details in Section 4.
==============================================================================
4. THE HEIGHT FIELD LIMIT
==============================================================================
4.1 THE FLAT-DOMAIN LIMIT.
Theorem 4.1 (flat-domain GFF limit). Let U_delta be either a single
deterministic flat chamber, or a finite mesh-dependent union of
independent chambers with pairwise disjoint interiors, all inside a
fixed disc B_0. Suppose F_delta := closure(B_0 \ U_delta) converges in
Hausdorff distance to a compact connected set F containing d B_0, and
put U = int(B_0 \ F). Then
K_delta => h_U / a in H^s_loc(U), s < -1,
where h_U is the zero-boundary GFF on U with covariance 2 pi (-Delta)^{-1}
in each component, and its restrictions to different components are
INDEPENDENT zero-boundary GFFs. All finite joint moments with supports
in U converge. The extended-by-zero fields K_delta are uniformly tight
in H^s_loc(B_0) for every s < -1.
Remark 4.1-i. "Different components even when their discrete
precursors were one chamber" is the content of 3.5: exterior
orthogonality of the parent chamber's outside data can hold even when
a single discrete chamber pinches into several limiting components.
The rest of Section 4 proves Theorem 4.1 in stages.
4.2 Limiting chambers and their Green kernels.
Lemma 4.2A (Green convergence). Let F_n be connected compact subsets
of closure(B_0) containing d B_0, with F_n -> F in Hausdorff. Put
V_n = B_0 \ F_n and V = B_0 \ F. Then every component of V_n and of V
is simply connected, every compact subset of V is eventually contained
in V_n, and
G_{V_n}(x, y) -> G_V(x, y) locally uniformly for distinct x, y in V,
including x, y in different components of V. The zero-extended Green
kernels satisfy 0 <= G_{V_n}, G_V <= G_{B_0} and are Borel in the
argument (F_n, x, y).
Proof. Simple closed curve complement fact (in the sphere, a simply
connected open set has connected complement and vice versa) plus
connectedness of F_n makes every chamber simply connected. Every
compact subset of V is eventually in V_n by Hausdorff convergence.
Green convergence is standard Caratheodory kernel convergence. Borel
measurability is automatic from the definition via Dirichlet energy
on a countable exhaustion. []
4.3 Logarithmic moment bound.
Proposition 4.3A (logarithmic moments). Let z_1, ..., z_{2m} be
corner-grid sites in a flat chamber of diameter <= R_0/2. Then
|E prod_{i=1}^{2m} K(z_i)| <= C_m sum over pairings P of
prod_{ {i, j} in P } (1 + log^+(R_0 / (|z_i - z_j| OR delta))).
Odd moments vanish. The same bound holds for independent flat
chambers after setting the field to zero off those chambers. For a
compactly supported bounded test f, uniformly over all such chambers,
E |K_delta(f)|^{2m} <= C_m ||f||_infty^{2m}.
For f_{r, z}(x) = r^{-2} f((x - z)/r), 0 < delta <= r,
E |K_delta(f_{r, z})|^{2m} <= C_{m, f} (1 + log^+(R_0/r))^m.
Sketch. Condition on the current N and the s spins. By Lemma 3.1
consequence (3), the centered height at a wall-parity site is a sum of
jumps of size 2 with independent fair signs, one sign per contributing
cluster. A cluster's coefficient at a site is zero or has magnitude 2.
Its sign is shared by all its odd holes. Summing over clusters and
using the cluster count tail 3.3D gives the display. The scaled version
follows from rescaling x = z + r u and from the delta/r factor being
essential for the atomic smearing: after the rescaling the diagonal
part has an additional uniformly bounded factor (delta/r)^2 log(r/delta),
while the common scale contributes 1 + log^+(R_0/r). []
4.4 Boundary decay.
Proposition 4.4A (boundary decay). Consider deterministic flat
chambers in a fixed bounded region with connected accessible walls,
and a Hausdorff limit of their complements. Fix a limiting open
component and a point in its interior. For a site within distance r of
the matching connected wall, and a site at distance >= R from that
site, a shared cluster joining the two height contributions has
probability at most C (r/R)^eta for some C, eta > 0, provided
delta <= r << R.
Proof. A shared cluster is either incident to the distinguished site
or encloses it. In either case its connected passage must avoid the
matching circuits attached to this wall at every separated scale
between r and R, since such a circuit separates the two contributions.
The conditional circuit bound 3.3B gives a uniformly positive per-
scale success probability. Iterating over the geometrically separated
scales gives C(r/R)^eta. []
Corollary 4.4B. Under the same hypotheses, the limiting moment kernel
T_j has zero boundary trace in each variable while the other variables
stay fixed at distinct interior points.
4.5 Tightness and subsequences.
Lemma 4.5A (flat tightness). With F_delta -> F in Hausdorff:
(1) The extended-by-zero fields K_delta are uniformly tight in
H^s_loc(B_0) for every s < -1.
(2) Along a subsequence of meshes, the fields converge in law to a
limit field R, and the j-th moment distributions T_j (as measures
on U^j) converge weakly. The limit extends to the compactified
domain closure(B_0 \ F).
(3) All finite joint moments of the field R against smooth tests of
compact support in U exist and are limits of the lattice moments.
Proof sketch. The uniform second-moment bound of 4.3A gives
sup_delta E || chi * K_delta ||_{H^{-q}}^2 <= C sum_{n in Z^2}
(1 + |n|^2)^{-q} < infinity
for q > 1, over a fixed torus tiling with normalised Fourier basis e_n.
For 1 < q < -s, compactness of H^{-q} -> H^s_loc gives tightness.
A countable exhaustion of cutoffs plus summable exceptional probabilities
gives tightness on the whole of B_0. Higher versions of 4.3A give
uniform integrability of all finite test products; weak convergence
follows. []
4.6 Small revealed sets.
Lemma 4.6A (centered remainders on small sets). Conditional on a
stopped exploration using 3.1, let R_delta be the centered flat
remainders set to zero on exposed sites. Let E_delta be a
stopped-measurable set of sites in a fixed bounded region, and let
m_delta be the area of the lattice cells of E_delta. If m_delta -> 0
in probability, then for every bounded deterministic test f,
R_delta(f * 1_{E_delta}) -> 0 in L^2.
The conclusion is uniform over stopped geometries. Additional revealed
means bounded by a constant also contribute vanishingly.
Proof. Condition on stopped data. The residual fields are independent
flat fields by 3.1. For every terminal-measurable union E of cells in
the test support, the logarithmic covariance bound gives
E |R_delta(f 1_E)|^2 <= C ||f||_infty^2
int int_{E x E} (1 + log^+(R_0 / (|z - w| OR delta))) dz dw
<= C' ||f||_infty^2 |E|,
the second inequality following from the uniform integral of the
logarithmic kernel on a bounded set. Averaging over E_delta and using
the convergence of E[m_delta] to zero in probability gives the
result. []
4.7 Harmonicity of limiting moment kernels.
Proposition 4.7A (harmonicity). Along a subsequence as in 4.5, the
limiting j-th moment distributions T_j admit locally smooth
representatives off the collision diagonals, and each such
representative is separately harmonic in each variable away from the
other collision points.
Proof. On a product of disjoint interior disks, truncate the spectator
product, apply 3.5 to each insertion variable in turn (with the other
variables' disks fixed and spectator tests fixed), and remove the
truncation with the higher moment bounds of 4.3A: this gives
Delta_{x_i} T_j = 0 in the distributional sense,
off the collision set. Summing these equations yields an elliptic
equation on the product space; hence a smooth separately-harmonic
representative. The tightness of 4.5 rules out additional
distributional mass supported on the collision diagonals. []
4.8 Logarithmic expansion at a collision.
Lemma 4.8A (uniform logarithmic expansion). Let (y, z) be a
collision-free list of interior points. Then there is a smooth
coefficient L_j(y, z) such that near x = y:
T_j(x, y, z) = L_j(y, z) log(1 / |x - y|) + R_j(x; y, z),
where R_j extends harmonically in x across y. On a compact set of
allowed (y, z), the expansion holds uniformly, and the remainder and
each fixed order of its x-derivatives are uniformly bounded on a
smaller common radius.
Proof. Standard elliptic estimate for separately harmonic functions
with logarithmically integrable growth at the collision, applied on
concentric annuli and patched with the maximum principle. []
4.9 The collision coefficient.
Proposition 4.9A (collision coefficient). For j >= 2 and distinct
(y, z) in U,
L_j(y, z) = kappa * T_{j-2}(z),
where kappa is the plane covariance coefficient of 3.4F.
Proof. Fix two smooth zero-mass tests alpha, beta with disjoint
supports and nontrivial log pairing
J = int int alpha(u) beta(v) log(1/|u-v|) du dv != 0.
For an explicit choice let rho_b be a smooth radial unit bump supported
within distance 1/10 of (b, 0), all rho_b obtained by translating one
fixed bump. Take alpha = rho_0 - rho_1 and beta = rho_3 - rho_4. Radial
harmonic averaging gives J = log(8/9) != 0.
Fix radial unit mollifiers in disjoint disks around the spectator
points z. By separate harmonicity (4.7A and 4.8A) the average of
T_{j-2} in those disks equals T_{j-2}(z). The same is true of T_j
when its first two points lie in a small disk around y disjoint from
the spectator disks. Apply 4.10A (neutral-pair mixing) and the
subsequential covariance 3.4F. For each fixed r, the plane pairing is
kappa * J up to an addable constant log(1/r) that vanishes against
neutral tests. So
lim_{r -> 0} int int alpha(u) beta(v)
T_j(y + r u, y + r v, z) du dv
= kappa J T_{j-2}(z).
There is no additional large-L limit in this choice of tests.
On the other hand insert the log expansion 4.8A in the left side. The
term L_j(y + r v, z) log(1/r) integrates to zero against alpha
EXACTLY, before any continuity estimate. The remaining term tends to
L_j(y, z) J uniformly in the spectator variables on compact sets. For
the remainder, subtract R_j(y; y + r v, z) inside the u-integral; the
subtracted term again integrates to zero, and the uniform first-
variable derivative bound gives an O(r) error. Comparing with the
previous display proves L_j = kappa * T_{j-2}. []
4.10 Wick recursion in a flat component.
Proposition 4.10A (Wick recursion). For every j >= 2 and distinct
points in U:
T_j(x_1, ..., x_j) = kappa sum_{i=2}^j G_U(x_1, x_i)
T_{j-2}(x_2, ..., hat{x_i}, ..., x_j).
Consequently the limiting centered field in U is Gaussian with
covariance kappa * G_U. In particular, restrictions to different
components are independent.
Proof. Fix x_2, ..., x_j and subtract the right side of the recursion
from T_j. As a function of x_1, the difference is separately harmonic
off the points x_i by 4.7A. The collision coefficient identity 4.9A
cancels every logarithmic pole in its own component, and 4.8A removes
the remaining punctures. Points in other components do not produce a
pole because G_U is zero-extended between components. The resulting
harmonic function has zero boundary trace by 4.4B, and the Green
functions have zero boundary trace by 4.2A. It is bounded near the
boundary, and its finitely many interior punctures are removable, so
by the maximum principle it is zero on each component. This proves the
recursion, including mixed moments when discrete components have
separated only in the limit. []
Corollary 4.10B. Starting from T_0 = 1 and T_1 = 0, the recursion
gives zero odd moments and the sum over perfect pairings of products
kappa G_U(x_{i_k}, x_{j_k}) for even moments. The products are
locally integrable. Since 4.5 excludes diagonal mass, the identity
holds for the smeared moment distributions. These moments determine
the law: for any finite linear combination Z of smeared fields with
variance s^2,
E Z^{2n} = (2n - 1)!! s^{2n}, E Z^{2n+1} = 0,
E |Z|^n <= s^n (2n)^{n/2}.
The exponential series is integrable for every real argument, so the
characteristic function is exp(-s^2 u^2 / 2). Apply to all finite
linear combinations and then to a countable determining family of
tests in the local Sobolev space. The zero cross-component covariance
proves independence. []
4.11 Plane increment kernels.
Proposition 4.11A (plane Gaussian correlations). Along the chosen
subsequence, all plane increment correlation functions Phi_{m, delta}
converge uniformly on compact subsets of the configuration space which
allows the two endpoints of a single increment to coincide. The limit
Q_m is zero for odd m, and for even m is the sum over pairings of
products of C_kappa.
Proof. Induct on m. The cases m = 0, 1, 2 are direct (m = 0 is trivial;
m = 1 vanishes by height reflection; m = 2 follows from 3.4F). For
even m >= 4, subtract from Q_m the sum over i = 2, ..., m of
C_kappa(d_1, d_i) Q_{m-2}(hat{d_1}, hat{d_i}).
In the variable x_1, the plane collision coefficient cancels every
logarithmic pole at x_i and y_i (i >= 2), and its harmonic remainders
remove the punctures. The regularity estimate of DKLM gives continuous
extension through x_1 = y_1 (value zero there; this is the collapse
of the first increment, not an additional collision between different
factors). The remainder is consequently entire harmonic. With all
other endpoints fixed, the same estimate bounds its growth by a fixed
power of 1 + log(2 + |x_1|), which is o(|x_1|). An entire harmonic
function of sublinear growth is constant: the interior gradient bound
on disks of radius M, followed by M -> infinity, makes its gradient
zero. At x_1 = y_1 both Q_m and the recursion sum vanish. The constant
is therefore zero. This proves the plane Wick recursion at every even
order. The quoted equicontinuity plus an increment-collapse estimate
extend the identity locally uniformly across the collapse set. []
4.12 Identifying the coefficient from free energy.
At this point we have along a chosen subsequence: plane increment
kernels are Gaussian with covariance C_kappa, and flat-domain limits
are Gaussian with covariance kappa * G_U. Kappa could be anything.
Proposition 4.12A (plane normalization). For every subsequence
considered above,
kappa = 1 / a^2,
where a^2 = 2 pi arcsin(c/2), equivalently a = sqrt(g) * lambda with
g = 8/pi arcsin(c/2) and lambda = pi/2.
Proof. The proof is logically sensitive.
(1) Once the covariance of the plane increment law is identified
as Gaussian with coefficient b_0 in the display of 3.4F, the free-
energy correspondence of DKLM 2026 (Lemma 8.4 in DKLM paper on the
six-vertex model) computes b_0 directly from the torus free energy.
The result is b_0 = 1/(2 pi arcsin(c/2)), hence kappa = b_0 in the
normalization used here.
(2) The height K_delta has log coefficient 1/(2 pi); the source GFF
h has covariance 2 pi (-Delta)^{-1}. Therefore sigma^2 (in the
notation of DKLM: the variance multiplier) = 1 / arcsin(c/2), and
kappa = sigma^2 / (2 pi) = 1 / (2 pi arcsin(c/2)) = 1 / a^2.
(3) The conversion to original height variable is via K_delta = 2 H_delta,
which gives H_delta -> h/(pi sqrt(g)).
The logical order is important. The spectral argument of 3.4 gave only
a subsequential coefficient. The local collision argument of 4.9 and
4.10 identified every Gaussian correlation. The convergence criterion
of DKLM (Theorem 4.5) then supplied the required input to the free-
energy correspondence. Only now has the free energy fixed the
coefficient. There is no circularity. []
[BOX] Order of operations, one more time, since this is the key
logical point. (a) Moment bounds give subsequences. (b) Reflection
positivity in two frames identifies the plane covariance as beta
times Green, with beta unknown. (c) Screening plus local mixing
produce harmonic moment kernels. (d) The logarithmic expansion at a
collision gives collision coefficients kappa T_{j-2}; the max principle
turns this into the Wick recursion. (e) Gaussianity follows from the
recursion. (f) DKLM's free-energy formula, whose input has just been
supplied by (e), pins down beta = 1/a^2.
4.13 Completion: proof of Theorem 4.1.
Proof of Theorem 4.1. By 4.2A every compact subset of U is eventually
inside the discrete open set. On a finite list of interior buffers,
4.5A gives joint subsequential field and moment convergence. The
screening identity 3.5 applies there with the fixed positive
clearances. The boundary estimate 4.4A uses the connected wall in
each chamber and holds at every boundary point of the limiting
components. The Green convergence 4.2A supplies the matching zero
trace. Thus the proofs of 4.9A and 4.10A apply to these components.
The coefficient identity 4.12A gives kappa = 1/a^2. The same argument
applies to each component separately with independent zero-boundary
fields. Multiply the characteristic functions of independent
components. []
4.14 Conditional law at a stopped cut.
Proposition 4.14A (conditional flat limit). Let E_delta be the
stopped sigma-field of a stopping exploration (3.1). Let F_delta be
the exposed closed set with its one-cell thickening, and let e_delta
be E_delta-measurable stopping data. Suppose for some fixed s < -1
along a subsequence
(e_delta, F_delta, R_delta) => (e, F, R) in E x K(B_0) x H^s_loc(B_0),
with E a Polish space and K(B_0) the compact subsets of closure(B_0)
in the Hausdorff metric. The limit F is a.s. connected and contains
d B_0. Set U = B_0 \ F and
A_F = { dist(union_i supp(f_i), F) > 0 }.
Then a.s.
A_F * E[ exp(i sum u_i R(f_i)) | sigma(e, F) ]
= A_F * exp( -(1/(2 a^2)) sum_{i, j} u_i u_j G_U(f_i, f_j) ).
Thus, conditional on sigma(e, F), the restrictions of R to the
components of U are independent zero-boundary GFFs divided by a,
including components born from a single pinched discrete precursor.
Proof. For a deterministic realized chamber configuration, the flat
product law supplies the characteristic function in the first line
of the display. The error from the displayed Gaussian characteristic
function tends to zero uniformly over all such configurations with
the stated clearance. Otherwise choose a sequence with the error
bounded away from zero. Compactness of closure(B_0) \ F in K(B_0)
gives a Hausdorff-convergent further subsequence, and the clearance
persists in the limit. The deterministic theorem 4.1 gives
convergence of the flat characteristic function to the Gaussian one
for that limit, and 4.2A gives G_{U_delta}(f_i, f_j) -> G_U(f_i, f_j)
on the same subsequence. The two characteristic functions therefore
have the same limit, contradiction. []
Theorem 4.1B (stopped-cut conditional GFF). Let U_delta, E_delta,
F_delta be as in 4.14A, with the residual product class of chambers
at the stopping time. Then the conditional characteristic functions
of the deviation R = h - M, where M is the stopped-measurable local
mean field (piecewise-constant label of the flat chamber), converge
uniformly over the stopped data to those of a zero-boundary GFF with
covariance 2 pi (-Delta_U)^{-1}, restricted to the components of U.
The mean field M is identified in Section 5.
Proof. Decompose h = M + R on the stopped configuration. The flat
label M is a stopping-measurable harmonic function off the exposed set
(by 3.1 consequence (2) and (3)); the centered remainder R is killed
on the exposed set by construction and zero-extension is an
H^1-isometric operation there. Then the conditional flat limit
4.14A applies to R with the constant label removed. The only point
that needs attention is that M is measurable with respect to the
stopped sigma-field; this is the identification theorem of
Section 5, which we use only as a black box here. []
==============================================================================
5. THE CURRENT-CLUSTER LIMIT
==============================================================================
[BOX] Headline of Section 5. Independently of the height limit, the
lattice clusters of both the wired and the free current converge to
canonical two-valued-set clusters of the SAME h that appears in
Theorem 4.1. The subtlety is that a lattice cluster is a set of edges,
while its continuum limit is a fractal boundary; the identification
goes through two operations: (i) stopped-carpets identify at each
finite odd depth; (ii) the partition of odd holes into shared clusters
is preserved through the limit, using the sign-correlation identity.
5.1 Standard two-valued sets, in one place.
We use the notation already defined in 1.6. If we want to be precise:
for a fixed zero-boundary GFF f on a Jordan domain O and constants
alpha, beta > 0 with alpha + beta >= 2 lambda, the canonical thin
two-valued set TVS_{alpha, beta}(f) is the unique field-measurable
closed set including d O such that: each component H of O \ TVS has
the conditional mean constant, either -alpha or +beta on H; the
remainder f restricted to H, after subtracting this mean, is an
independent zero-boundary GFF on H; and d O is always included. This
is ASW 2017 Proposition 2. Monotonicity, nested structure, and Jordan
property of complementary loops follow (ASW 2017 Sections 3-4).
Special case: TVS_{2 lambda, 2 lambda} has CLE_4 complementary loops.
5.2 Stopped two-valued sets converge.
Proposition 5.A (stopped two-valued sets, Prop 3.4 + 3.5 of original).
Let ({C_partial_delta}) be the first wired layer; let ({C_delta^r}) be
the carpet at finite odd depth r as in Definition 1.8; let ({F_delta^r})
be the global closed carpet at finite odd depth r. Along the joint
subsequence of Theorem 4.1,
(1) The wired boundary cluster C_partial_delta converges in Hausdorff to
TVS_{a,a}(h).
(2) In a free flat parent with incoming flat value b, and after
subtracting b, the first odd carpet F_delta^1 converges in Hausdorff
to TVS_{2a, 2a}(h^{parent}), where h^{parent} is the free-parent
zero-boundary GFF remainder.
(3) For each finite odd depth r, the nested closed carpet F_delta^r
converges in Hausdorff to the canonical nested carpet F^r of h.
Labels and restarted zero-boundary fields converge on every finite
family of pointed compact buffers in the limiting holes.
Proof sketch. The field converges by 4.1. The stop at fixed finite
depth is a stopping-time local set (Prop 3.2 of the original). The
conditional field in each sealed chamber is a flat GFF plus a constant
label (by 3.1 consequence (2)). At the wired layer, the difference
between the initial boundary value and the boundary cluster is +-a;
bounded-type thin local set uniqueness (ASW 2017 Prop 2) identifies
TVS_{a,a}(h). At the first odd layer, a monochromatic W perimeter of
the CLE_4 holes forces a split identity; the bounded-type theorem
identifies the whole closed carpet with TVS_{2a,2a}(f). Induction on
odd depth r uses 4.1B in each hole after subtracting the label. Finite
depth rules out pathologies by taking limits first at fixed depth.
The given full formal proof (Section 7.x of the original) fills in the
cable GFF test of no-crossing, the label recovery, and the area
vanishing as in Lemma 3.3C and Prop 3.1. []
5.3 Nested completion.
Lemma 5.B (nested completion, Lem 7.9 of original). Let f be a
zero-boundary GFF in a bounded simply connected parent P. Let E be a
field-measurable boundary-connected bounded thin local set with value
c_H constant on each component H of P \ E, with
-p <= c_H <= q, p, q > 0, p + q >= 2 lambda,
and all c_H in a fixed deterministic countable set L subset [-p, q].
For each component H with -p < c_H < q, insert the relative set
TVS_{p+c_H, q-c_H} in its zero-boundary remainder; leave components
already labelled -p or q untouched. Then the closed union of the
inserted sets equals TVS_{p, q}(f).
Proof sketch. The inserted set has bounded-density mean field (the
density is bounded by 2 lambda on every test). Local-set theory
(ASW 2017 Thm 2.3(3)) makes the closed union a local set. Each
complementary component has a constant mean of -p or q by construction,
so the union is a thin boundary-connected local set with those flat
values; uniqueness as in 5.1 gives TVS_{p,q}. [] For full details,
see ASW 2017 Sections 3-4 and the original Section 7.
5.4 The label of the stopped field.
Lemma 5.C (recovering the flat label, Lem 9.11 of original). Consider
a stopped exploration at a seed stop or at a fixed odd-depth carpet in
a flat chamber. The stopped-measurable local mean M_delta (piecewise
constant on exposed cells) is a.e. a constant distribution on each
interior buffer of every unexposed open component. Its pointwise value
equals the flat label of the enclosing chamber (up to additive
constants on exposed sites).
Proof sketch. On any compact set at positive clearance from the frozen
seed and the initial exposed data, the centered field is a zero-boundary
flat GFF with covariance as in Theorem 4.1B. Take the decomposition
h = M + R with M as in 4.1B, with R centered and harmonic mean zero.
Testing against bump functions: M(f) is harmonic, bounded on the
compact set, and satisfies a mean-value identity from the GFF's own
harmonicity, so it must equal the constant labelling. This constant
extends continuously to the exposed set because the clearance
conditions used in 4.1B hold uniformly. []
5.5 No crossing.
Proposition 5.D (no-crossing, Prop 8.1 of original). Let O be a
chamber of reachable component at a finished seed stop. Let W_0 be
the initial boundary of the revealed support. Let C* be a canonical
cluster of h (either free or the boundary layer). Then C* cannot
simultaneously meet O and intersect W_0 inside the domain with
positive diameter. More strongly: if C* has positive-diameter
intersection with both O and the boundary of O (in the trace topology
of the stopped cut), the intersection of C* with the closed complement
of O must have vanishing diameter.
Proof sketch. Suppose a positive-diameter piece C* touches both.
Then by Lemma 3.5F (flat GFF on the sealed side), the conditional mean
of the insertion M restricted to a buffered path test between the two
touches cannot be flat. The cable GFF interior-exhaustion lemma
(Lem 8.6 of original) makes it possible to choose a compactly
supported test whose support lies entirely inside O and whose bulk
overlaps C* at positive diameter. Apply the reverse buffer estimate
(Lem 8.3 of the original) and the local kernel identification
(Lem 8.4) to reach the conditional mean from OUTSIDE the stopped cut.
The stopped-cut mean is the same on both sides of the parameter test.
A diametrically opposite pair of such tests then contradicts 3.3C
(the collar) uniformly in the clearance. []
[BOX] Why "no-crossing"? A continuum cluster cannot be both inside
and outside the stop boundary. If a lattice cluster is subdivided by
the stop boundary into two macroscopic pieces, the pieces must
converge to DIFFERENT canonical clusters, not the same one. This is
what makes the "one discrete cluster -> one canonical cluster"
direction of the partition argument possible.
5.6 The partition of odd holes.
Proposition 5.E (visible partition, Prop 9.9 of original). Let R be a
fixed finite collection of marked odd holes. Let eps_delta(i) be the
fair sign of the discrete current cluster supplying the hole V_delta(i)
(with the convention eps_delta(i) = 0 if V_delta(i) is not a valid
hole). Let eps(i) be the corresponding canonical fair sign. Then along
the joint subsequence of Theorem 4.1 and 5.A:
(1) eps_delta(i) -> eps(i) in law.
(2) The relation i ~_delta j ("V_delta(i) and V_delta(j) are odd holes
of ONE discrete cluster") converges in probability to i ~ j
("V(i) and V(j) are odd holes of ONE canonical cluster").
Proof. Part (1). By Theorem 5.A the observed hole V_delta(i) converges
in Hausdorff. Condition on the unlabelled data (N, s). By Lemma 3.1
(3), the fair sign eps_delta(i) is independent of (N, s). Its signs
agree across all odd holes of one cluster. The fair-sign statement of
Lemma canonical-geometry (the continuum analogue) then gives the same
fairness for eps(i), and 5.A transports the marked-hole limits. So (1)
holds.
Part (2). Key identity. Conditional on (N, s), the signs eps_delta are
independent fair on distinct clusters, hence
E[eps_delta(i) eps_delta(j)] = P(i ~_delta j).
On the continuum side, conditioning on the unlabelled recursive
geometry, signs on distinct canonical clusters are independent fair
(Lem canonical-geometry, last paragraph), hence
E[eps(i) eps(j)] = P(i ~ j).
Passing to the joint limit, the L^2-convergent product of eps_delta's
converges to the product of eps's, so the left-hand sides agree in the
limit. Therefore
P(i ~_delta j) -> P(i ~ j).
Partition inclusion. By 5.D (no-crossing), a visible discrete cluster
has connected positive-diameter limit; if it supplied BOTH hole i and
hole j, then both V(i) and V(j) lie inside the SAME canonical cluster
(by the "one discrete cluster -> one canonical cluster" direction, which
is the maximum principle applied in 4.14A with the thinness of the
canonical family). So "i ~_delta j" implies "i ~ j" eventually.
Combining with the previous display gives the equality of probabilities
with the lattice and the continuum relations. []
Corollary 5.E' (all-generation local finiteness, Cor 9.10 of original).
Almost surely, the entire canonical free cluster family contains only
finitely many members of diameter greater than any eps > 0. Same for
the wired family including all descendants.
Proof. Fix eps > 0. Consider M canonical free clusters of diameter
greater than eps. In each cluster choose two distinct rim points at
distance > eps/2 (possible by density of odd rims in the canonical
cluster, a consequence of thinness and the definition of the recursion).
By Prop 5.E each such pair of witnesses is realised by a distinct
discrete precursor of diameter > eps/4 for small enough mesh. So M
canonical clusters of diameter > eps imply M lattice clusters of
diameter > eps/4 for large enough mesh. Counting by 3.3D with
union bound and Fatou makes this event tend to zero as M -> inf. []
5.7 Complete matching.
Proposition 5.F (complete matching, Prop 9.17 of original). Let
(H_delta, C^w_delta, C^f_delta) be the joint process on the chosen
subsequence (or on any full-measure subsequence). Then the following
hold simultaneously almost surely.
(a) C^w_delta converges (in the matching topology of Section 1.9) to
the wired family C^w = "wired boundary cluster TVS_{a,a}(h),
followed by the free recursion in each component of D \ TVS_{a,a}(h)".
(b) C^f_delta converges to the free family C^f = F_a(h, D).
(c) The distinguished boundary component C_partial_delta converges in
Hausdorff metric to TVS_{a,a}(h).
(d) The joint limit is unchanged if we discard a countable union of
null sets; in particular, the fields and both collections are all
deterministic functions of the same h.
Proof outline. Use Prop 5.A to identify the wired boundary and finite-
depth carpets with the canonical objects. Use Prop 5.E pair by pair to
identify which odd holes belong to one canonical cluster at every
finite depth. Use Cor 5.E' to establish local finiteness of the canonical
family at all generations, hence that the limit collections are closed
sets in the matching topology. Together (5.A + 5.E + local finiteness)
identify the support limit of the discrete family with the canonical
family.
The additional ingredients are:
(i) exclusion of "invisible duplicate precursors". A discrete cluster
has a positive-diameter Hausdorff limit C*. If C* is not the boundary
cluster C_partial and is not one of the recorded visible canonical
clusters along the SAME subsequence, we derive a contradiction as
follows. Choose an axial direction e in which the projection of C*
has positive width w. The recorded stops at ranks according to
direction e with buffer b = w/4 stabilise for all sufficiently fine
meshes. One fixed rank supplies the same C* along infinitely many
meshes; replacing the original sequence by this fixed-rank sequence
loses nothing about the support limit. The fixed-rank sequence is one
of the "visible" sequences counted by 5.E, hence has the canonical
cluster as its limit. This contradicts the "not visible" hypothesis.
(ii) pointwise thinning of individual holes. Even when the closed
carpet converges, the closures of individual holes may fail to
converge pointwise; but pointed convergence is enough, because the
canonical family is determined by the pointed odd-hole data (Cor 5.E').
(iii) multiplicities. If a discrete family has two clusters with the
same positive-diameter canonical limit C*, the matching topology
discards one of them (Section 1.9); we have therefore established
matching convergence to C^w and C^f as set-valued objects, which is
exactly the conclusion of Theorem 2.1.
Combining these steps gives matching convergence and joint
measurability. []
==============================================================================
6. ASSEMBLY: PROOF OF THE MAIN THEOREM
==============================================================================
6.1 The two halves combine.
Proof of Theorem 2.1. Run a simultaneous subsequence along which all
of the following hold:
(a) K_delta -> h/a in H^s_loc(D), by 4.1 and 4.5A.
(b) The stopping explorations used to expose the boundary cluster
and the free family in the joint limit are captured by 5.A and
5.F, with all generations of the canonical family realised as
limits.
(c) The free dual family is captured by 5.F.
(d) The conditional law of the field in stopped chambers is the
one described in 4.1B and 5.C.
Such subsequences exist because:
- 4.5A gives tightness of the field in H^s_loc and tightness of
the moment distributions along subsequences;
- 5.A gives joint subsequential limits for wired and carpet stops
at every finite odd depth;
- 5.F gives joint limits for the complete cluster families on that
same subsequence, with the matching topology as the topological
space.
Since every subsequence on which (a)-(d) hold gives the same joint law
of (h, C^w, C^f, C_partial) by the canonical construction of Section 1.7,
and since the joint laws on any two such subsequences agree by
uniqueness of the pushforward under the canonical realisation,
the full mesh limit exists.
The height convergence to H = K/2 = h/(2a) = h/(pi sqrt(g)) is exact.
6.2 Measurability of the collection-valued maps.
The construction C^f = F_a(h, D) and C^w = (TVS_{a,a}(h), F_a of the
remainder in every component) defines a pair of maps h -> (C^w, C^f)
into the matching space of Section 1.9. These maps are measurable:
enumerate recursive clusters by generation and by the lexicographically
first mark of a fixed dense test family Q in each generating hole,
using a cemetery value for missing entries; the cluster-residual
map is Borel by Cor 5.E; append the boundary cluster TVS_{a,a}(h) as
one extra slot for the wired family. Cor 5.E' shows that the
singleton-augmented family is closed; every open V in the hyperspace
gives a hit event which is a countable union of cluster hit events.
Hence the maps are Borel measurable. See the original Section 9.8 for
the precise Borel structure.
6.3 Tightness.
Joint tightness of (H_delta, C^w_delta, C^f_delta, C_partial_delta) as
delta -> 0 follows from 4.5A (height tightness) plus the fact that the
cluster maps are measurable and defined on the same probability space
by the joint exploration of Section 5. Concretely, tightness of the
height in H^s_loc implies tightness of the field-measurable maps C^w,
C^f, C_partial, since these are deterministic functions of the field
on a set of probability one.
6.4 The theorem, in full.
Combining 4.1 (flat-domain GFF limit for the height), 5.A (wired
boundary and finite-depth carpets), 5.F (complete matching families),
5.E' (local finiteness at all generations), and 6.1-6.3 (compactness
and measurability), we have:
(i) H_delta -> h/(pi sqrt(g)) in H^s_loc(D) for every s < -1;
(ii) C_partial_delta -> TVS_{a,a}(h) in Hausdorff;
(iii) C^w_delta -> C^w and C^f_delta -> C^f in the matching topology;
(iv) All these convergences are joint on one single coupling of the
same field h.
This is exactly the statement of Theorem 2.1.
6.5 The Potts endpoint.
The same proof covers a = 2 lambda, i.e. the four-state Potts endpoint.
In this case the FPS decomposition still holds (ASW 2018 Thm 3 in the
version cited), the two-valued set TVS_{2a, 2a} is well defined, and
the bounded-type uniqueness theorem of 5.1 still applies. The area
vanishing lemma used in the free iteration (Prop 3.1 of the original)
has the endpoint limit p(z) = 1/2 at a = 2 lambda, and the argument
proceeds unchanged. []
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REFERENCES, IN PLAIN ENGLISH
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For the reader who wants to check sources:
[1] J. Ashkin, E. Teller (1943). The original four-component lattice
model. Where the name comes from.
[2] M. Lis (2022), "Spins, percolation and height functions".
Lattice-to-current correspondence and the finite-volume current
weight (EW). This is the basis of Lemma 3.1.
[3] M. Lis (2022), "On boundary correlations in planar Ashkin-Teller
models". The wired and free boundary conditional distributions
and the six-vertex identification.
[4] H. Duminil-Copin, M. Lis, W. Qian (2025, versioned arXiv). The
double random current case; gives an analogous result for an
Ising point of a related model; not the same as ours.
[5] H. Duminil-Copin, K. Kozlowski, D. Lammers, A. Manolescu
(versioned 2026). Six-vertex model: height convergence for
sqrt(3) <= c <= 2, extended inputs on 1 <= c <= 2 (log moment
bounds, percolation circuits, free-energy correspondence). This
is the analytic basis of Theorem 3.3A, Input 3.4-iv, Input 3.4-i,
and Prop 4.12A.
[6] J. Aru, T. Lupu, A. Sepulveda (2020, 2018 versioned, 2023
versioned). Two-valued local sets, first-passage sets, excursion
decomposition. This supplies the canonical two-valued sets and
their uniqueness (5.1), and the CLE_4 structure.
[7] S. Alcalde Lopez, K. Heeney, M. Lis (2026, versioned). The
conjecture we prove (Conjecture 1.2), the topology (Section 1.9),
and the Jordan-boundary version of the joint limit (Thm 1.14).
[8] S. Alcalde Lopez, A. Sepulveda (2026, versioned). A stronger
joint decomposition at the Ising point U = 0 (g = 2). Not
required for our theorem.
[9] W. Sierpinski (1918). A continuous image of a compact connected
set is either one point or uncountably many points. We use the
form "a countable disjoint closed cover of a connected compact
set has at most one nonempty member" (Section 5, main text).
[10] R. C. Freiwald (2014), "An Introduction to Set Theory and
Topology", Chapter 5. Clean reference for the Sierpinski
theorem.
[11] L. V. Ahlfors (1979). Complex Analysis (3rd edition). Caratheodory
kernel convergence, conformal Green function facts used in
Section 4.
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