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 Fechel-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