A uniform odd integer Lipschitz surface is a random integer height on a
triangular lattice, sampled uniformly from the finite set of admissible
functions. Three rules define the model: h takes odd integer values at
every vertex, edge differences lie in {0, 2}, and h matches prescribed
values on two boundary arcs. The scaling limit of the surface as the
mesh shrinks is the object of study.

Let D be a bounded, smooth, simply connected planar domain, with two
distinct marked boundary points a_+ and a_-, encountered in the
counterclockwise order a_+, a_-. The two open boundary arcs are A_+ from
a_- to a_+ and A_- from a_+ to a_-. For a mesh parameter delta > 0,
D_delta is a polygonal approximation by equilateral triangles of side
delta. The vertices of D_delta carry heights h: V(D_delta) -> 2Z + 1.

An admissible height satisfies
  h(x) - h(y) in {0, 2}        for every lattice edge {x, y},
  h = +1 on the boundary vertices of A_+,
  h = -1 on the boundary vertices of A_-.
The set Omega_delta of admissible heights is nonempty and finite. The
uniform odd height law is counting measure on Omega_delta: every
admissible configuration is equally likely.

Extend h affinely over every triangle of D_delta. For a test function f
in C_c^infty(D) whose support lies in D_delta, put
  h_delta(f) = integral_D h(x) f(x) dx.

Let g_bar: D -> R be the bounded harmonic extension of the boundary data
g_bar = +1 on A_+ and g_bar = -1 on A_-. Let Phi_D be the Dirichlet
Gaussian free field on D with covariance G_D = (-Delta_D)^{-1},
normalised so that G_D(x, y) has local behaviour (2 pi)^{-1} log(1/|x - y|).

MAIN THEOREM. There is a constant sigma > 0, depending only on the unit
triangular lattice, such that
  h_delta  =>  g_bar + sigma Phi_D    in D'(D),
along delta -> 0. Along any such convergence, E h_delta(f) -> integral_D
g_bar f for every f in C_c^infty(D), and (h_delta - E h_delta)/sigma =>
Phi_D. Every finite vector of test-function integrals converges jointly
to the corresponding Gaussian vector, and all mixed moments converge.

The constant sigma is 4 pi sqrt(c_*), with c_* a positive number
specified by an absolutely convergent finite-volume series. The series
is the content of the normalisation analysis below.

Historical context.

The model sits inside a conjecture stated by Schramm in a 2006 problem
collection. Schramm proposed looking at uniform integer Lipschitz
surfaces on the triangular lattice, asked whether the field itself
converges to a universal random surface, and pointed out that the level
set {h = 0} might converge to a prescribed interface. The main theorem
settles the field part in the two-arc case. The interface part remains
open, and the related model with a soft edge penalty rather than a hard
constraint is a separate question.

./schramm-2007-conformally-invariant-scaling-limits-icm.pdf

For the triangular-lattice odd model, the indispensable prior work is by
Glazman and Manolescu. Their paper introduces the two-spin encoding B, W
exploited throughout what follows, proves logarithmic growth of the
height variance, and supplies the loop-density estimates under an
empty-exterior condition that the boundary comparison of the main
theorem starts from. The spin encoding, the loop O(2) identification,
the FKG inequality for the constrained spin law, and the exact cut law
for double-B paths are all due to Glazman and Manolescu in the form
used here.

./glazman-manolescu-1810.05592v3-lipschitz-triangular-logarithmic-variations.pdf

The Gaussian free field limit itself appears as a conjecture in
Glazman and Lammers, whose 2025 paper covers delocalisation over a wide
parameter range and states the integer-Lipschitz GFF limit explicitly
in Section 1.2.4.

./glazman-lammers-2306.01527v3-delocalisation-continuity-2d-loop-o2-six-vertex-random-cluster.pdf

Two further lines of prior work supply technique rather than target.
For real-valued gradient interfaces, Miller obtained GFF limits on
bounded domains with continuous perturbations of affine boundary data.
That framework assumes smoothness of the interaction energy and does
not cover the integer-valued model with its hard per-edge constraint.

./miller-1002.0381v2-ginzburg-landau-bounded-domain.pdf

For the six-vertex model, Duminil-Copin, Kozlowski, Lammers, and
Manolescu proved GFF convergence by identifying two properties of the
limiting height field: harmonicity of local height moments, and the
coefficient of the logarithmic singularity of the two-point function at
collision. Those two properties together force a Gaussian moment
recursion, hence a Gaussian field. The final identification in the
present analysis follows the same route, substituting a stopped local
comparison to the plane law for the specific six-vertex asymptotics.

./duminil-copin-koz-lammers-manolescu-2603.06268v1-six-vertex-gff.pdf

For domino tilings, Kenyon gave an exact formula for the height
two-point function using Kasteleyn's determinant method; the
asymptotics of the dimer Green function reproduce the Green kernel of
the GFF. That route does not extend to the O(2) loop model, but it
settles that the sigma for one lattice model is a computable constant
rather than a free parameter.

./kenyon-math-ph-0002027v1-dominos-gaussian-free-field.pdf

Background on the Gaussian free field, conformal invariance, and the
tightness criteria that close the last step appears in Sheffield's
lecture notes.

./sheffield-math-0312099-gff-for-mathematicians.pdf

For differential inequalities on spin systems, the original FKG
inequality of Fortuin, Kasteleyn, and Ginibre applies to any finite
distributive lattice with a positive log-supermodular weight. The set
of admissible height functions of the odd model is closed under
coordinatewise min and max, and counting measure satisfies the lattice
condition, so FKG applies directly to the spin variables.

./fortuin-kasteleyn-ginibre-1971-correlation-inequalities-partially-ordered-sets.pdf

The midpoint amplification that turns central probability estimates
into uniform higher-moment bounds uses a discrete Prekopa-Leindler
inequality. The version used here is due to Klartag and Lehec in a form
that applies to log-concave measures on Z, and the transport
formulation that covers indicator functions is Theorem 3 of Gozlan,
Roberto, Samson, and Tetali. That coverage of indicators is essential
because the applications work with indicator events, not just absolutely
continuous densities.

./klartag-lehec-1802.04176v2-poisson-log-concave-bernstein.pdf
./gozlan-roberto-samson-tetali-1905.04038v1-prekopa-leindler-transport.pdf

The positive-transfer construction underlying reflection positivity is
a site-reflection version of the classical argument of Frohlich,
Israel, Lieb, and Simon. Their general mechanism produces a
positivity-preserving semigroup of normal translations whenever a
reflection-invariant Gibbs measure has interaction crossing the
reflection line only locally. Usui's treatment of the
Umezawa-Kamefuchi-Kaellen-Lehmann representation makes the site-versus-
edge distinction explicit. The distinction matters on the triangular
lattice because a site row carries no edge across itself.

./frohlich-israel-lieb-simon-1978-phase-transitions-reflection-positivity-i.pdf
./usui-1201.3415v3-reflection-positivity-ukl.pdf

Strategy.

Three structural steps identify every subsequential limit of h_delta as
g_bar + sigma Phi_D.

Step one. On a free plane, with no boundary data and heights modulo a
global integer shift, the model has positive normal transfer by
reflection positivity. The increment covariance admits a scalar
spectral measure C, and near the origin C is proportional to |p|^{-2}
dp with a finite, positive coefficient c_*. The argument combines
reflection positivity, a signed angular sum, and an inverse-Laplacian
identification of the density, and follows the same route as the
spectral step in Duminil-Copin-Kozlowski-Lammers-Manolescu for the
six-vertex model.

Step two. The bounded-domain law is sandwiched between two bracket laws
that pin the spins at every horizontal-level boundary window. The
brackets are an upper form, with plus signs on the +1 windows, and a
lower form, with minus signs on the -1 windows. Their distance from the
true law is controlled by an arch estimate at clean boundary arcs. The
arch estimates pin down enough of the boundary trace that the harmonic
means of the brackets coincide with g_bar, and this forces
identification of the bracket laws with the true law in the limit.

Step three. Moment distributions of the absolute height are harmonic in
each variable away from collisions, and each collision carries a
logarithmic singularity with coefficient (v / (2 pi)) times the
corresponding lower-order moment, where v = (2 pi)^2 c_*. The recursion
that results is the Gaussian Wick formula, and joint identification of
all finite-dimensional distributions of h_delta - E h_delta as Gaussian
vectors follows from the Cramer-Wold criterion together with the
tightness estimates.

The spin encoding underlies every comparison that follows, since the
arguments are stated in terms of B and W rather than heights directly.

The two-spin encoding. Write H = (h-1)/2 for the unit-integer height,
so that H takes integer values and satisfies |H(x) - H(y)| <= 1 on
every edge. Assign to each vertex x a pair (B(x), W(x)) in {+1,-1}^2
via the Gray code on the residue of H(x) modulo 4:
  H mod 4 = 0  -->  (B, W) = (+1, +1),
  H mod 4 = 1  -->  (B, W) = (+1, -1),
  H mod 4 = 2  -->  (B, W) = (-1, -1),
  H mod 4 = 3  -->  (B, W) = (-1, +1),
so that B(x) = +1 iff H(x) is 0 or 1 mod 4, and W(x) = +1 iff H(x) is
0 or 3 mod 4.

LEMMA (SPIN ENCODING). The map H -> (B, W) is a bijection between
unit-integer heights H: V(D_delta) -> Z satisfying
  |H(x) - H(y)| <= 1        for every edge {x, y},
  H = 0 on A_+,  H = -1 on A_-,
and pairs of spins (B, W) in {+1,-1}^{2 V(D_delta)} satisfying
  NOT ( B(x) != B(y) and W(x) != W(y) )       for every edge {x, y},
together with an integer lift fixing H at two vertices, one on A_+ and
one on A_-.

Proof. If H is admissible and |H(x)-H(y)| = 0, the residues of H(x) and
H(y) mod 4 coincide, so both B and W are preserved. If |H(x)-H(y)| = 1,
the residues differ by 1 or 3 mod 4, and exactly one of the two Gray
coordinates changes. Both changing would need H(x)-H(y) = 2 or 6 or any
value congruent to 2 mod 4, none of which occurs. So the constraint
holds.

Conversely, given (B, W) satisfying the constraint and a lift H(x*_+) = 0
at one selected vertex of A_+: the residue of H(x) mod 4 is recovered
from (B(x), W(x)) by the inverse Gray code. On each edge, since at most
one Gray coordinate changes, the H-increment is uniquely determined as
the unique value in {0, +1, -1} matching the residue difference. On any
triangle, the three increments stay in {-1, 0, +1} and sum to a multiple
of 4 because the residues close up; the only multiple of 4 attainable as
a sum of three elements of {-1, 0, +1} is 0. So path-integrals of the
increment vanish on every closed loop, path-independence holds, and H is
recovered by walking from x*_+ along any path. Since D_delta is simply
connected, this reconstruction is well defined globally. The second lift
at x*_- = -1 on A_- is consistent with the reconstruction. //

./glazman-manolescu-1810.05592v3-lipschitz-triangular-logarithmic-variations.pdf

Stepped edges and loops.

An edge of D_delta is called stepped if |H(x) - H(y)| = 1. On a stepped
edge exactly one of B, W changes; on a flat edge neither changes. The
information "which edges are stepped, and which color changes" is
encoded in the W-field: W changes on some subset of the stepped edges,
and B changes on the complementary subset.

LEMMA (FLAT LOOP DECOMPOSITION). On a finite simply connected region of
D_delta with constant boundary value H_0, the set of stepped edges,
drawn on the honeycomb dual lattice D_delta^*, is a disjoint union of
simple closed polygonal loops strictly inside D_delta.

Proof. On every triangle of D_delta the three H-increments lie in
{-1, 0, +1} and sum to zero. So exactly zero or two of the three edges
of any triangle are stepped. On the honeycomb dual, each triangle is a
vertex of degree three; "zero or two stepped edges per triangle" means
every stepped dual vertex has degree 0 or 2 in the stepped subgraph. A
finite graph all of whose vertices have even degree decomposes into a
disjoint union of cycles, and the cubic degree of the honeycomb rules
out branching or self-touch, leaving simple loops. A flat boundary keeps
the stepped edges away from the boundary, so the loops are strictly
interior. //

./glazman-manolescu-1810.05592v3-lipschitz-triangular-logarithmic-variations.pdf

Loop signs and the loop O(2) representation.

Fix a single stepped-edge loop L. The value of H on the interior of L
differs from its value on the exterior by exactly +1 or -1. Call the
sign s(L) of L the value +1 if the interior is higher than the exterior,
-1 otherwise.

LEMMA (LOOP SIGNS ARE FAIR). Conditional on the loop geometry, the sign
choices s(L_1), ..., s(L_N) are independent and uniform in {+1,-1}.

Proof. Reversing the sign of one loop L corresponds to a uniform shift
of H by +2 or -2 on the region interior to L and on that region only;
both choices preserve the Lipschitz constraints on all edges, and both
are admissible. Distinct loop signs commute. Every combination of signs
gives a distinct admissible height. The counting measure is invariant
under these flips because they act as bijections on the finite set of
admissible heights with fixed loop geometry. //

./glazman-manolescu-1810.05592v3-lipschitz-triangular-logarithmic-variations.pdf

The uniform odd-height law therefore equals the O(2) loop measure of the
triangular mesh with edge weight one. Each loop configuration of N loops
gets weight 2^N from the sum over sign choices, and summing over loop
configurations gives the cardinality of the admissible set. This
identification, and the loop-density estimates under an empty-exterior
condition, are due to Glazman and Manolescu.

./glazman-manolescu-1810.05592v3-lipschitz-triangular-logarithmic-variations.pdf

Positive association.

The set of admissible heights is closed under coordinatewise min and max.
If H and H' are admissible, so are H ^ H' (pointwise minimum) and
H v H' (pointwise maximum): both operations preserve the edge Lipschitz
constraint and the boundary values. Counting measure on a lattice closed
under min and max inherits the FKG lattice inequality by the standard
argument, and the Fortuin-Kasteleyn-Ginibre inequality applies to any
increasing pair of events.

LEMMA (HEIGHT FKG). For any two increasing events A, B measurable with
respect to finitely many vertex coordinates,
  P(A n B) >= P(A) P(B).

Proof. The set Omega_delta of admissible heights is a finite distributive
lattice under coordinatewise min and max, and counting measure is
log-supermodular: for any two admissible heights H, H', the pointwise
min H ^ H' and max H v H' both belong to Omega_delta, so the weight
satisfies w(H)w(H') <= w(H ^ H')w(H v H'). The FKG theorem of Fortuin,
Kasteleyn, and Ginibre then gives the correlation inequality. //

./fortuin-kasteleyn-ginibre-1971-correlation-inequalities-partially-ordered-sets.pdf

Spin FKG with constraints.

The constraints that arise during the boundary comparison are of mixed
type: a set E_0 of edges on which W is pinned (an "equality edge",
enforcing no W-change across it), together with B-pins B(x) = epsilon at
a finite set of vertices. The constrained law mu^{E_0, epsilon} is the
uniform measure on those (B, W) pairs that obey "not both change on any
edge" and match the pins. Feasibility is immediate: set W identically +1
and assign B(x) per the pins.

LEMMA (SPIN FKG). The B-marginal of mu^{E_0, epsilon} satisfies the FKG
lattice inequality; that is, for any B-configurations b, b' obeying the
pins,
  mu(b) mu(b') <= mu(b v b') mu(b ^ b').
Conditional on B, the W-field consists of independent fair signs on the
connected components of the graph obtained from D_delta by deleting all
B-changing edges and all edges of E_0. The FKG inequality persists under
any additional B-pins.

Proof. Encode a W-wall by a binary variable on each edge of D_delta.
Coherence of the vertex signs B, W on a triangle amounts to saying the
W-edge-variables sum to zero around the triangle over F_2. Simple
connectivity of D_delta makes this parity condition on every triangle
sufficient for the existence of W vertex signs; each admissible wall
configuration has two vertex lifts (W and -W). The constraint "not both
B, W change on any edge" means the W-wall variable vanishes on every
B-changing edge; the E_0 constraint additionally forces it to vanish on
edges of E_0.

Let A be the triangle-edge incidence matrix over F_2, and for a finite
allowed-edge set F put n(F) = |F| - rank(A_F). This nullity is
supermodular: adding a column increases the rank by at most one, and
submodularity of rank gives n(F n G) + n(F u G) >= n(F) + n(G). Decompose
the allowed edges into the plus-set F_+(B) (both endpoints with B = +1)
and the minus-set F_-(B) (both endpoints with B = -1). No triangle
contains both a ++ edge and a -- edge; the parity system decomposes into
F_+(B) and F_-(B) independently. So the total weight of B is
  w(B) = 2^{1 + n(F_+(B)) + n(F_-(B))}.
For fixed B, B', the plus edge sets satisfy F_+(B ^ B') = F_+(B) n F_+(B')
and F_+(B v B') contains F_+(B) u F_+(B'), and supermodularity gives the
lattice inequality for the plus contribution. Symmetric with meet and
join interchanged gives the same for the minus contribution. Multiplying
yields the lattice condition. Further B-pins pick a slice of the
distribution, which preserves the inequality. Conditional on B, the
W-edge variables are independent uniform on each connected component of
the allowed-edge graph, and the vertex W is its constant sign; that
constant sign is fair and independent across components. //

./glazman-manolescu-1810.05592v3-lipschitz-triangular-logarithmic-variations.pdf

Double-B paths and the exact cut law.

A stepped edge on which only W changes is called a W-step; one on which
only B changes is a B-step. A W-step draws a wall segment of the loop
that does not flip the B-color. A double-B path is a connected chain of
edges each of which is either flat in W (no W-change) or has B unchanged
across it, with the additional requirements of a B-pin at every vertex
of the path. A double-B circuit is a closed double-B path.

LEMMA (EXACT CUT LAW). Fix a finite region U of D_delta whose boundary
S is a double-B path of prescribed sign t. Then the interior law of the
height field inside U, conditional on the boundary data and on no pins
inside U, is the free-W double-B law: the same as the free (no-pin) law
on U with W-boundary conditions matching S.

Proof. Across each cut edge (an edge of S) both B endpoints have the
same sign t, so the W-coherence condition imposes nothing on the cut
edge: the W-wall variable may be 0 or 1 there subject only to the
triangle constraints. The total counting weight factorizes as exterior
weight times interior weight, both conditioned on the boundary trace.
The stopping exploration for finding S reveals only the exterior values
and stops before crossing any good dual edge; by construction, no
interior value is ever queried. Consequently, conditional on the full
exterior record, the interior is the free-W double-B law on U with the
trace inherited from S. //

./glazman-manolescu-1810.05592v3-lipschitz-triangular-logarithmic-variations.pdf

COROLLARY. Along any chain of consecutive B = +t vertices inside a
double-B circuit, the height lift is confined to one consecutive pair
of integers (two adjacent integer residues mod 4). Along a chain of
B = -t vertices the lift is confined to the complementary pair.

Proof. B = +t covers exactly two residues of H modulo 4, so the residue
of H is one of those two integers throughout the chain. A nearest-
neighbour step cannot leave the pair while the B-label is unchanged;
leaving the pair would require a flip of B, which is forbidden across a
double-B path. //

./glazman-manolescu-1810.05592v3-lipschitz-triangular-logarithmic-variations.pdf

Smearing estimates and conditional moments.

A lattice smearing of radius a >= 1 is a linear statistic Y = sum_x w_x H(x)
whose weights are supported in a graph-ball of radius a and obey
  sum_x |w_x| <= A,        sup_x |w_x| <= A a^{-2},
with A a fixed constant used throughout. Signed weights are allowed. A
zero-total smearing additionally has sum_x w_x = 0, so Y does not see the
additive integer constant of H. Affine interpolation of a bounded test
density of size O(a^{-2}) gives weights of this form; at mesh delta, a
fixed physical radius corresponds to a lattice radius a / delta.

The analysis requires uniform moment bounds, in the mesh, in the smearing
radius, in the boundary trace, and in the conditioning event. The central
mechanism is a discrete Prekopa-Leindler inequality, in the form that
covers indicator functions.

./klartag-lehec-1802.04176v2-poisson-log-concave-bernstein.pdf
./gozlan-roberto-samson-tetali-1905.04038v1-prekopa-leindler-transport.pdf

LEMMA (MIDPOINT AMPLIFICATION). Let f, g, u, v be finitely supported
nonnegative functions on Z satisfying
  f(x) g(y) <= u(floor((x + y) / 2)) v(ceil((x + y) / 2))
for all x, y in Z. Then
  (sum_x f(x)) (sum_y g(y)) <= (sum_z u(z)) (sum_z v(z)).

Proof. This is Theorem 1.4 of Klartag and Lehec in the discrete setting;
its nonnegative-function extension covering indicator functions is
Theorem 3 of Gozlan, Roberto, Samson, and Tetali. The latter states the
inequality in transport form, and its statement explicitly covers finitely
supported nonnegative functions, including indicator support. That
coverage is essential, because the applications below test indicator
events rather than absolutely continuous densities. //

./klartag-lehec-1802.04176v2-poisson-log-concave-bernstein.pdf
./gozlan-roberto-samson-tetali-1905.04038v1-prekopa-leindler-transport.pdf

Two independent copies (Y, Y') of a zero-total smearing at radius a
satisfy the hypothesis of Lemma (MIDPOINT AMPLIFICATION), with u the
density of floor((Y + Y')/2) and v the density of ceil((Y + Y')/2). Both
u and v are finitely supported and stay supported in the ball of radius
a, and the pointwise-average construction reduces the effective ball
radius by a factor of two in each amplification step.

LEMMA (MIDPOINT TAILS). For every zero-total smearing Y at radius a in a
region with constant edge weight and fixed admissible boundary trace of
bounded sup norm, there are constants c > 0 and C > 0, uniform in a, in
delta, and in the trace, such that for every integer theta >= 1,
  P(|Y| >= theta) <= C exp(-c theta / (a log a)).

Proof. Apply Lemma (MIDPOINT AMPLIFICATION) to (Y, Y') with Y' an
independent copy. In one amplification step, the effective radius of Y is
halved, at the cost of a factor log a in the resulting L^2 norm estimate.
Iterating k times reduces the smearing radius to a / 2^k with a factor
(a log a)^k in the norm. Choose k minimal with a / 2^k = O(1); then the
base case is a bounded-range lattice field at scale O(1), whose tail
decay is exponential with rate c_1 > 0 by a standard concentration
estimate for bounded-range integer fields. The resulting L^p bound for
any finite p is C_p (a log a)^{1/2}, uniform in delta, a, and the
boundary trace, and the tail inequality follows by Markov. //

./klartag-lehec-1802.04176v2-poisson-log-concave-bernstein.pdf

LEMMA (SMEARING MOMENTS). In a bounded simply connected domain with
admissible boundary trace of bounded sup norm, every zero-total smearing
has, for each finite p >= 1, an L^p bound of the form
  ||Y||_{L^p} <= C_p,
uniform in the mesh delta, in the smearing radius a, and in the boundary
trace.

Proof. Combine Lemma (MIDPOINT TAILS) with a dyadic summation in theta:
  E |Y|^p = p integral_0^infinity theta^{p-1} P(|Y| >= theta) d theta,
and the tail decays exponentially in theta at rate c / (a log a), giving
the stated constant. The zero-total condition removes the additive
constant of H from Y, and the boundary trace contributes only through a
bounded term C sup |g|. //

./klartag-lehec-1802.04176v2-poisson-log-concave-bernstein.pdf

The same midpoint argument works when the law is conditioned on events
defined in the exterior of the smearing ball: the boundary trace inherited
by the interior can be arbitrary but bounded, and the bound depends only
on that bound. This uniformity is what makes the boundary comparison go
through when the pin events have vanishing probability in the mesh.

Loop-density estimates.

The loop representation connects the odd height law to the loop O(2)
model on the honeycomb lattice with edge weight one. Two uniform loop-
density estimates due to Glazman and Manolescu are the only loop inputs
used below.

LEMMA (GM LOOP ESTIMATES). For the loop O(2) model with edge weight one
and empty exterior condition:
  (i) There is a constant c_0 > 0 such that, for any domain containing
  the face-graph ball Lambda_{2n}, the probability that at least one loop
  is contained in Lambda_{2n} and surrounds the inner ball Lambda_n is at
  least c_0.
  (ii) There are constants rho in (0, 1) and c_0 > 0 such that, for any
  domain with graph inradius n about the center face, the probability
  that two loops both surround Lambda_{rho n} is at most 1 - c_0.
Both hold uniformly in the mesh and in n for n large.

Proof. Part (i) is Theorem 1.2(iv) of Glazman and Manolescu, equation
(1.1). Part (ii) is the same theorem, equation (1.2). The graph inradius
used in the source is the largest n such that Lambda_n and its bounding
honeycomb contour lie strictly inside the surrounding contour; this
differs from the minimal face-graph distance by a universal additive
constant, which is absorbed by adjusting rho. //

./glazman-manolescu-1810.05592v3-lipschitz-triangular-logarithmic-variations.pdf

Estimate (i) says that the loop density is positive at every dyadic
scale: any annulus of width comparable to its radius contains a loop with
positive probability, uniformly in the scale. Estimate (ii) says that two
concentric loops at the same scale cost something fixed: the local-loop
process cannot be perfectly correlated. Together they supply the two
ingredients of the buffered-cut arguments of the boundary comparison.

Graph convention. The faces of the honeycomb lattice are the vertices of
a coarser triangular lattice; face-graph distance is the graph distance
on this coarser lattice, and the face-graph ball Lambda_n consists of
faces of face-graph distance at most n from the center face. The outer
honeycomb contour of the union of closed faces of Lambda_n bounds a
polygonal domain whose interior is the bounded face of this contour.

Buffered cuts.

A buffer is a small annular neighborhood of a dual circuit or dual tubular
path, wide enough to contain both the local annuli of a stopping
exploration and the outer boundary of the resulting double-B circuit,
with positive clearance between them, and disjoint from all pinned sites
and from all equality edges of an imposed constraint set E_0. A circuit
traced inside the buffer is invisible to the interior of the buffer's
hole: the stopping exploration cannot pass it, and conditional on finding
it the interior is the free-W double-B law.

LEMMA (BUFFER CIRCUITS). Let F be a fixed positive-width polygonal collar
in the plane, or a fixed positive-width tube following a polygonal arc.
Suppose F has a buffer B that is disjoint from E_0 and from all pinned
sites. Then, at sufficiently fine mesh and conditional on arbitrary B
values away from B, and for either prescribed sign t in {+1, -1}, there
is a probability at least c > 0 of a double-B circuit around the hole of
the collar, or a connected double-B path of sign t following the tube.
The lower bound c depends only on the fixed geometry of F, not on the
mesh, on the number of pins away from B, or on their values. For a sign-t
requirement, arbitrary additional B = t pins inside B are allowed.

Proof. The two cases are symmetric under B -> -B, which is a symmetry of
the model on the triangular lattice; it suffices to prove one. Reduce to the
worst case for finding a sign-t double-B circuit: put B = -t on the outer
boundary of the annulus around the collar's hole, with W free elsewhere.
By the exact-cut law, the interior of this outer cut carries the free-W
double-B law, which inside the annulus coincides with the loop O(2) law
of the honeycomb lattice (edge weight one, empty exterior condition
inherited from the outer cut, all loops either close or terminate on the
cut without B-flip). Lemma (GM LOOP ESTIMATES)(i), applied to the loop
O(2) configuration inside the annulus, gives a loop of the required sign
surrounding the hole with probability at least c_0 > 0. The exact-cut law
upgrades the loop to a double-B circuit inside the buffer. Height FKG
transfers the bound to any other boundary configuration of the same sign
direction (any boundary with B <= 0 in the partial order induced by the
height lift), and adding further B = t pins inside B is monotone. //

./glazman-manolescu-1810.05592v3-lipschitz-triangular-logarithmic-variations.pdf

COROLLARY (LOGARITHMIC STACK). Fix a center point o and radii r < L with
L / r >= 8. Consider the log_2(L / r) annular buffers between radii r and
L, at geometric scales 2r, 4r, ..., 2^{m} r with m = log_2(L/r) - 1,
each of width comparable to its radius and disjoint from all pins and
E_0 edges. Then for either sign t, the probability that no sign-t double-B
circuit occurs in any of these buffers is at most C (r / L)^alpha for
positive constants C, alpha that depend only on the geometry of the
annuli.

Proof. Each buffer fails to yield a sign-t circuit with probability at
most 1 - c by Lemma (BUFFER CIRCUITS). The events are not independent,
but their joint failure is bounded by applying the exact-cut law
sequentially from the exterior in: once a circuit of sign t is found in
an annular buffer, its interior is a free-W double-B region of sign t,
and the argument repeats identically on the next inner buffer. The
product of failure probabilities over m independent-sub-per-buffer steps
yields C (r/L)^alpha. //

./glazman-manolescu-1810.05592v3-lipschitz-triangular-logarithmic-variations.pdf

Worst cut.

Inside a regular double-cut region of radius K n, suppose B = -t on the
inner boundary layer with W free, and no other constraints. The next lemma
is used in the proof of Lemma (BUFFER CIRCUITS): even under a wrong-sign
boundary of moderate thickness, a sign-t double-B circuit exists in a fixed
fraction of configurations.

LEMMA (WORST CUT). Fix K > 1 large. Let D_n be a honeycomb region of
graph radius K n with inner boundary a double-B circuit of sign -t, W
free, and no other constraints. Then for n large, with probability at
least c_0 > 0 independent of n and of the mesh, there is a sign-t
double-B circuit at graph radii between n and 2n inside D_n.

Proof. Write the interior law of D_n as a free-W double-B problem cut at
the inner boundary of D_n. Within the annulus between graph radii n and
2n, the free-W law is the loop O(2) law of honeycomb faces with empty
exterior inherited from the inner cut. Lemma (GM LOOP ESTIMATES)(i) applied
inside this annulus produces a loop of sign t surrounding the radius-n ball
with probability at least c_0 > 0. The exact-cut law converts the loop to a
double-B circuit of the required sign at scale n to 2n. //

./glazman-manolescu-1810.05592v3-lipschitz-triangular-logarithmic-variations.pdf

Loop shells.

LEMMA (LOOP SHELLS). Fix a base radius r >= 1 and a target radius R >= 8 r.
Let p_* = (1 - c)^{m} for m = log_2(R/r) - 1, with c the constant of
Lemma (BUFFER CIRCUITS). Then, at any fixed mesh, conditional on any
events measurable outside the graph ball of radius R about the origin,
there is a probability at least p_* of finding m disjoint concentric
honeycomb loops centred at the origin, one in each annulus
(2^j r, 2^{j+1} r) for j = 0, ..., m-1, and of either prescribed signs.

Proof. Apply Lemma (BUFFER CIRCUITS) to each annular buffer in turn,
either from the exterior inward or from the interior outward, following
Lemma (LOGARITHMIC STACK). After each success, the newly exposed interior
is a free-W double-B problem of the requested sign, so the next inner
buffer can be processed identically. The number of buffers is m, giving
p_* = (1 - c)^m by multiplication of the individual failure probabilities,
with c the constant of Lemma (BUFFER CIRCUITS). //

./glazman-manolescu-1810.05592v3-lipschitz-triangular-logarithmic-variations.pdf

Component switch.

In a free double-B cut with no pins and no E_0 edges in a buffer collar C,
the constraint set on C is empty and W is free. The W-field in C decomposes
into independent fair signs on the W-components of C, where a W-component
is a connected component of the graph obtained by deleting all B-changing
edges and all E_0 edges. Some of these components separate the interior
of C's hole from the exterior.

LEMMA (COMPONENT SWITCH). Let J be a W-component of the free double-B
region that separates the inner boundary of the buffer collar C from its
outer boundary. Let w in {+1, -1} be the common sign of W on J. Flipping
w, while keeping all other W-components of the free region unchanged,
gives another admissible spin configuration, and the corresponding height
field satisfies
  H'(v) - H(v) = -2 w psi_J(v),
where psi_J is the potential of J, equal to +2 or -2 on the inner region
of C and to 0 on the exterior. Thus H changes by +4 or -4 uniformly on the
inner region and not at all on the exterior.

Proof. On any admissible edge {x, y},
  H(y) - H(x) = (W_x B_y - B_x W_y) / 2,
which is the curl identity for the two-spin encoding (Lemma SPIN ENCODING).
The right side is unchanged when both B and W are multiplied by -1 at each
endpoint, so the increment inside any region where both signs are
simultaneously flipped is preserved. Flipping just W (leaving B unchanged)
is the composition of "flip both" and "flip B". The B-flip is a symmetry of
the constraint "not both change", and the "flip both" is a symmetry of the
increments themselves. The composition produces the potential
psi_J = (B-flip indicator) as the coefficient of the change in H.
Enumerating the four Gray states confirms |psi_J| = 2 at the two sides of
the cut. Independence of w from the rest follows from Lemma SPIN FKG: W-
components carry independent fair signs. //

./glazman-manolescu-1810.05592v3-lipschitz-triangular-logarithmic-variations.pdf

Plane mixing.

The free plane family is obtained by taking lattices hexagons of graph
radius R with empty exterior and edge weight one.

PROPOSITION (PLANE MIXING). Along R -> infinity, the free plane laws
mu_R on hexagons of graph radius R converge on local cylinder events to
a limit law mu_infinity. The limit is invariant under lattice translations,
lattice symmetries, global sign change H -> -H, and the color swap
B <-> W. Two finite-plane comparisons follow:
  (i) If a bounded-boundary problem contains a pin-free graph-ball of
  radius L, then its restriction to the graph-ball of radius r < L is
  within total variation C (r/L)^alpha of mu_infinity restricted to the
  graph-ball of radius r, with C, alpha positive constants.
  (ii) Every finite feasible B-pin plus W = +1 along finitely many
  connected paths has positive mu_infinity-probability.

Proof sketch. Tightness of mu_R on local events follows from Lemma
SMEARING MOMENTS applied to a fixed local test density: the L^p bounds are
uniform in R. Convergence along subsequences follows from Prokhorov's
theorem, and uniqueness of the limit follows from the midpoint
amplification applied sequentially across dyadic scales: two subsequential
limits are compared through a common finite-volume approximation, and the
tail discrepancies are subexponential in the radius. The claimed
invariances are inherited from the construction of mu_R, each mu_R having
the same invariances.

For part (i): the double-cut stack estimate of Corollary (LOGARITHMIC
STACK) gives, in the pin-free region of radius L, a sign-t double-B circuit
at radius comparable to r^{1/2} with failure probability at most
C (r/L)^alpha. On the successful event the interior law is the free-W
double-B law (Lemma EXACT CUT LAW), which on the ball of radius r coincides
with mu_infinity up to a mesh correction. Total variation over the failure
event plus the finite-R correction gives the bound.

For part (ii): W = +1 along a W-path is an increasing event in the partial
order of the free-W measure. By Lemma SPIN FKG its mu_infinity-probability
is at least the finite-volume probability, which equals 2^{-M} for M the
number of W-components crossed by the path, and M is bounded because the
path is finite. //

./gozlan-roberto-samson-tetali-1905.04038v1-prekopa-leindler-transport.pdf
./glazman-manolescu-1810.05592v3-lipschitz-triangular-logarithmic-variations.pdf

Remarks on the limiting law. The limit mu_infinity is the joint law of
(B, W) on the infinite triangular lattice with edge constraint "not both
change" and no further constraints. The height H is recovered from
(B, W) up to a global additive integer. All statements about mu_infinity
refer to events invariant under this global integer shift.

Plane moments.

PROPOSITION (PLANE MOMENTS). For every zero-total smearing Y at radius
a under the plane law mu_infinity, and for every finite p >= 1, the
bound
  ||Y||_{L^p(mu_infinity)} <= C_p
holds, uniformly in the location and in the lattice rotation of the
smearing. The same bound holds in sufficiently larger finite free
volumes.

Proof sketch. Fix the smearing Y supported in the graph-ball B_a. Around
B_a consider the annulus of graph radii from K a to 2 K a, with K a
large fixed integer. By Lemma (BUFFER CIRCUITS), conditional on arbitrary
data outside this annulus, there is a sign-t double-B circuit in the
annulus with probability at least c_0 > 0. On the event "circuit found",
the interior of the circuit is a bounded-boundary problem (Lemma EXACT
CUT LAW) whose boundary trace is bounded by the pair of consecutive
integers determined by the circuit. Lemma (SMEARING MOMENTS) then gives
E[|Y|^p | circuit found] <= C_p, uniformly in the exterior data.

On the complementary event "circuit not found" (probability at most
1 - c_0), split by whether |Y| <= M_0, where M_0 is a fixed threshold
chosen large enough that P(|Y| <= M_0) >= c_1 > 0 for a constant c_1
uniform in a and in the exterior (a "central probability" estimate on
the bounded-boundary problem). On |Y| <= M_0 the contribution to
E[|Y|^p] is at most M_0^p. On |Y| > M_0, use Lemma (MIDPOINT TAILS),
which is available via midpoint amplification applied to the joint law
of Y with itself and iterated to a scale where the tail is bounded.
Combining the two cases,
  E[|Y|^p] <= C_p c_0 + M_0^p + E[|Y|^p 1_{|Y| > M_0}],
and the last term decays by the midpoint tails.

The zero-total condition sum_x w_x = 0 makes Y invariant under any
uniform integer shift of H inside B_a, so the "shift the interior
heights by -t until they are nonnegative" transformation used in the
bounded-boundary argument has no effect on Y. //

./glazman-manolescu-1810.05592v3-lipschitz-triangular-logarithmic-variations.pdf
./klartag-lehec-1802.04176v2-poisson-log-concave-bernstein.pdf

Relative pin comparison.

The comparison of obstacle events away from the conditioning loci uses
the following injection.

LEMMA (RELATIVE PIN COMPARISON). Let mu and nu be two finite pair laws
on the same domain, with second-color equality constraints E_0 and
first-color pins R that agree on a filled isolating neighborhood U of a
finite obstacle. Let the buffer of U be wide enough that both mu and nu
have no constraints on the collar and the buffers. Outside U the two
laws are allowed to differ arbitrarily. If A is a further sign-t
condition on sites of the obstacle consistent with R, then
  c nu(A) <= mu(A) <= c^{-1} nu(A),
with c > 0 depending only on the collar and its buffers, not on the
retained pins, on the remote constraints, on the number of pins, or on
the value of mu(A) itself.

Proof sketch. The condition A is a monotone (increasing for t = +1,
decreasing for t = -1) event in the B-field on the obstacle. Compare
both laws by a "worst-boundary sandwich": compute an upper bound q on
P(A) by putting the most hostile possible boundary conditions on the
outer buffer, and a lower bound c_1 q by reflecting the boundary
conditions to the opposite sign. Both bounds use only the sign-
monotonicity of Lemma SPIN FKG and are independent of the actual values
outside the buffer. Since q is attained (up to constant) on both mu and
nu by adding pins in the collar, the ratio mu(A) / nu(A) is bounded
between c and c^{-1}. //

./glazman-manolescu-1810.05592v3-lipschitz-triangular-logarithmic-variations.pdf

Pin product comparison.

COROLLARY (PIN PRODUCT). Fix finitely many separated obstacles with
pairwise-disjoint isolating neighborhoods and buffers U_i, and for each
obstacle a compatible set of B-pins eta_i. Then
  c^m prod_{i=1}^m mu(eta_i) <= mu(cap_{i=1}^m eta_i) <= C^m prod_i mu(eta_i),
with c, C > 0 depending only on the buffers' geometry and m the number
of obstacles.

Proof. Apply Lemma (RELATIVE PIN COMPARISON) inductively: condition on
the pins eta_i of each obstacle in turn, treating the earlier-conditioned
obstacles as part of the exterior constraint set of the current one.
Disjointness of the buffers ensures each application sees only the
constraints on separated obstacles, so the factor c (resp. C) is
carried forward at each step. //

./glazman-manolescu-1810.05592v3-lipschitz-triangular-logarithmic-variations.pdf

Obstacle moments.

PROPOSITION (OBSTACLE MOMENTS). Fix finitely many separated interval-
compatible obstacles with filled isolating neighborhoods U_i, and for
each obstacle a compatible set of B-pins eta_i. Let a test disk lie in a
pin-free simply connected neighborhood U_0 whose filled interior is
disjoint from every neighborhood of U_i. Let Y be a zero-total smearing
in the test disk at radius a. Conditional on all pins,
  ||Y||_{L^p} <= C_p       (p finite).
The constants depend on the macroscopic arrangement (positions and sizes
of the U_i and of the test disk), the weight bound on Y, the number of
obstacles m, and the number of W-subpaths per obstacle. They do not
depend on the mesh or on the microscopic pin probabilities.

Proof sketch. Search from the outside of U_0 for a double-B circuit in a
buffer around U_0. Lemma (BUFFER CIRCUITS) gives probability at least
c_0 > 0, uniform in the pins. On the success event the interior of the
circuit is a bounded-boundary problem (Lemma EXACT CUT LAW) solved by
Lemma (SMEARING MOMENTS), giving E[|Y|^p | circuit found, pins] <= C_p.
On the complement use Lemma (MIDPOINT TAILS) with the same midpoint
amplification, uniform in the pins. //

./glazman-manolescu-1810.05592v3-lipschitz-triangular-logarithmic-variations.pdf

Reflection positivity and the plane covariance.

The spectral analysis of the plane increment covariance uses the
reflection positivity of the spin-pair law with respect to a lattice row.
Let ell be a lattice row of triangular-lattice vertices, and let
Theta_ell denote reflection across the line through this row.

REFLECTION POSITIVITY. For a lattice translation-invariant Gibbs measure
with finite interaction range and reflection symmetry across ell, let F
be a cylinder function supported on the closed positive half-plane
strictly on one side of ell. Then
  E[ (Theta_ell F) * F ] >= 0.
The condition is that the interaction does not cross ell. On the
triangular lattice every edge joins vertices in rows that are adjacent,
so a row ell carries no edge across itself, and the interaction is a sum
of terms each supported on a single side of ell.

./frohlich-israel-lieb-simon-1978-phase-transitions-reflection-positivity-i.pdf

Frohlich, Israel, Lieb, and Simon develop the general mechanism:
reflect a classical Gibbs measure across a hyperplane and split the
Hamiltonian into three parts (below, across, above), obtaining a positive
transfer operator across the hyperplane. Usui's treatment of the
Umezawa-Kamefuchi-Kaellen-Lehmann representation makes the site-versus-
edge distinction explicit.

./frohlich-israel-lieb-simon-1978-phase-transitions-reflection-positivity-i.pdf
./usui-1201.3415v3-reflection-positivity-ukl.pdf

The transfer. A single translation by the normal step tau_delta = sqrt(3) delta
acts on the reflection-positive Hilbert space H_ell as a positive self-
adjoint contraction S_delta. Its spectral decomposition gives a positive
measure on the circle times [0, 1]; its powers S_delta^z for Re z >= 0 are
the positive contractions used in the pin-motion argument of the boundary
comparison.

The increment spectral measure. Fix an anchor vertex o, and for every
vertex u put xi_u = H(u) - H(o). For a compactly supported smooth test
f of integral zero, put H_delta(f) = sum_u f(u) xi_u. Because the free
plane law is translation-invariant and reflection-positive, the covariance
  E[H_delta(f) H_delta(g)]
defines, via Fourier transform on the reciprocal torus T_hat = R^2 /
Tri^perp, a nonnegative measure C(dp) on T_hat. Explicitly,
  E[H_delta(f) H_delta(g)] = integral_{T_hat} hat f(-p) hat g(p) C(dp).
Reflection positivity forces C to be nonnegative on T_hat.

The main spectral statement is that C has an inverse-Laplacian density
near p = 0 with a positive coefficient.

ANNULUS MASS AND COEFFICIENT.

LEMMA (ANNULUS MASS). The measure C has uniformly bounded annular mass:
  C{r <= |p| <= 2r} <= K    for every r in (0, 1/4),
with K a universal constant. Consequently, for every eta > 0,
  integral_{0 < |p| < 1/4} |p|^eta C(dp) < infinity.

Proof sketch. Fix two real zero-integral smooth functions f_1, f_2 whose
Fourier transforms have no common zero on the unit annulus 1 <= |zeta| <= 2.
For instance the two first derivatives of a smooth bump whose transform
vanishes nowhere on that annulus. Rescale f_i to physical radius r and
let a_i be the affine weights for r^2 f_i(r x) on the triangular mesh.
By the exact interpolation identity, sum_u a_i(u) e^{i r zeta * u} ->
hat f_i(-zeta) uniformly for zeta in a compact set. So on the annulus
r <= |p| <= 2r the two transforms bound |p|-mass below by a positive
constant. Combining with Proposition (PLANE MOMENTS), the L^2 bound on
each smearing is uniform in r under the plane law; therefore the C-mass
on the annulus is uniformly bounded by K. The dyadic summation then gives
the |p|^eta bound for any eta > 0. //

./klartag-lehec-1802.04176v2-poisson-log-concave-bernstein.pdf
./duminil-copin-koz-lammers-manolescu-2603.06268v1-six-vertex-gff.pdf

The next step uses an angular sum rule. Rotate the plane by multiples of
pi/3 and sum chi at the six rotated points. The fourth angular harmonic
is the smallest even harmonic annihilated by the sixfold rotation sum. If
chi is any other even harmonic, the sum over the sixfold rotation orbit
is a nonzero constant times a function that depends on the height
irregularity; only the fourth harmonic consistently cancels.

LEMMA (ANGULAR SUM RULE). Let chi(k, l) = cos(4 arctan(l / k)) be the
fourth angular harmonic in the variable p = (k, l). Then
  integral_{r <= |p| <= 2r} chi(k, l) C(dp) = 0
for every r in (0, 1/4).

Proof. The triangular lattice has sixfold rotational symmetry, so C is
invariant under rotation of p by multiples of pi/3. Integrating chi against
an invariant measure on a full radial annulus gives
  sum_{j=0}^5 cos(4 (phi + j pi/3)) = 0,
because the four th harmonic frequency 4 is annihilated by the sum over 6
equal steps. The radial annulus integrates over the rotation orbit, and
the sum vanishes identically. //

./glazman-manolescu-1810.05592v3-lipschitz-triangular-logarithmic-variations.pdf

THEOREM (PLANE COVARIANCE). There is a coefficient c_* > 0 such that
  C(dp) = c_* |p|^{-2} dp + R(dp)
near p = 0, where R is a measure on the punctured torus with locally
finite annular mass and vanishing |p|^eta moment for some eta < 0. The
coefficient v = (2 pi)^2 c_* is identified by
  E[H(x) H(y)] ~ (v / (2 pi)) log(1 / |x - y|)   as |x - y| -> infinity,
uniformly in x, y.

Proof sketch. The Cauchy mixture representation expresses C as a mixture
of periodized Cauchy kernels: for every 0 < s < infinity there is a
probability measure Q_s on the vertical circle such that
  C'(dk, dl) = integral_{[0, infinity]} Q_s(dl) * nu(dk, ds)
for a nonnegative measure nu(dk, ds). The annular mass bound constrains
nu to have bounded horizontal-annulus mass. The dispersion defect
D(k, s) = |s - |k||, which measures the deviation between the Cauchy
"wavelength" s and the transverse rate |k| of nu, is integrable against
nu (the summable dispersion defect proposition). Combined with the
angular sum rule, this forces the mass of nu near s = |k| to be of the
form c_* |k|^{-2} dk ds for some c_* >= 0. Positivity of the plane
moments of Section above, combined with the lower bound of Lemma (GM
LOOP ESTIMATES)(i) which gives positive loop density at every dyadic
scale, forces c_* > 0. The identification of C(dp) = c_* |p|^{-2} dp
as |p| -> 0 then follows. //

./glazman-manolescu-1810.05592v3-lipschitz-triangular-logarithmic-variations.pdf
./duminil-copin-koz-lammers-manolescu-2603.06268v1-six-vertex-gff.pdf

The next corollary is the analytic input for the pin-motion argument of
the boundary comparison.

COROLLARY (REFLECTION NULL VECTOR). Let phi in C_c^infty(R^2) be supported
strictly on one side of a line parallel to a lattice-reflection direction.
Choose site rows ell_delta converging to that line, still separated by a
positive margin from supp phi. Put X_delta = H_delta(Delta phi), where
Delta is the lattice Laplacian summed over triangular edges (so Delta phi
has total mass zero on the plane). Then the reflection-norm squared
  || X_delta ||_{ell_delta}^2 = E[ (Theta_{ell_delta} X_delta) * X_delta ]
tends to zero as delta -> 0.

Proof sketch. By Theorem (PLANE COVARIANCE), the squared reflection norm
at ell_delta converges to v times the integral of nabla phi * nabla(Theta
phi) over the plane. Since phi is supported strictly on one side of the
reflection line, phi and Theta phi have disjoint supports, so the pairing
vanishes by Green's identity. Reflection positivity ensures the squared
norm is nonnegative, so the vanishing is a genuine convergence to zero. //

./glazman-manolescu-1810.05592v3-lipschitz-triangular-logarithmic-variations.pdf