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. Assign to every vertex x the residue of h(x)
modulo 4, translated via the Gray code
  h mod 4 = 1  -->  (B, W) = (+1, +1),
  h mod 4 = 3  -->  (B, W) = (+1, -1),
  h mod 4 = -1 -->  (B, W) = (-1, +1),
  h mod 4 = -3 -->  (B, W) = (-1, -1),
so that B(x) = +1 iff h(x) is 1 or 3 mod 4, and W(x) = +1 iff h(x)
is 1 or -1 mod 4.