arXiv subjects

Marianna Russkikh

Publications and source records attributed to Marianna Russkikh.

11 recordsLinked to original sources

arXiv2026

Harmonic functions on Tutte embeddings and linearized Monge-Ampère equation

We prove convergence of solutions of Dirichlet problems and Green's functions on Tutte harmonic embeddings to those of the linearized Monge--Ampère equation $\mathcal{L}_φh=0$. More precisely, we assume that piecewise linear Maxwell--Cremona potentials associated with the embeddings converge to a continuous potential $φ$ and the only assumption that we use is the uniform convexity of $φ$ or, equivalently, the uniform ellipticity of the operator $\mathcal{L}_φ$. Even if $φ$ is quadratic, this setup significantly generalizes known results for discrete harmonic functions on orthodiagonal tilings. Motivated by potential applications to the analysis of 2d lattice models on irregular graphs, we also study the situation in which the limits are harmonic in a different complex structure.

math-ph↗

arXiv2025

Perfect t-embeddings of doubly periodic Aztec diamonds

We study the large-scale geometry of t-surfaces -- pairs of perfect t-embeddings and their associated origami maps -- arising from dimer models on Aztec diamonds with periodic edge weights. We prove that these t-surfaces converge to space-like maximal surfaces in the Minkowski space $\mathbb{R}^{2,2}$. We observe that the frozen and gas regions influence the geometry of the limiting surface in striking ways: all frozen regions collapse to four boundary points, regardless of the number of frozen regions, while each gas region collapses to a distinct light-like cusp in the interior of the surface. In the absence of gas regions, the limiting surface lies entirely within $\mathbb{R}^{2,1}$; in the general case, however, this is no longer true. The limiting surface is sensitive to the detailed structure of the model: both the positions of the cusps, and the placement of the boundary vertices, depend on the precise way the edge weights are distributed on the Aztec diamond. Nevertheless, we show that the global conformal structure remains robust and coincides with the Kenyon-Okounkov conformal structure. We further conjecture that the cusp locations encode the shift in the discrete Gaussian component that appears in the global fluctuations of the dimer model.

math.PR↗

arXiv2025

Perfect t-embeddings and the octahedron equation of the two-periodic Aztec diamond

This paper explores the connection between perfect t-embeddings and the octahedron equation in the setting of the two-periodic Aztec diamond. In particular, we show that the positions of both the t-embedding and the corresponding origami map can be expressed as sums of density functions arising from solutions to the octahedron equation with appropriate flat initial conditions.

math-ph↗

arXiv2024

Perfect t-embeddings and Lozenge Tilings

We construct perfect t-embeddings for regular hexagons of the hexagonal lattice, providing the first example, and hence proving existence, for graphs with an outer face of degree greater than four. The construction is in terms of the inverse Kasteleyn matrix and relies only on symmetries of the graph. Using known formulas for the inverse Kasteleyn matrix, we derive exact contour integral formulas for these embeddings and their origami maps. Through steepest descent analysis, we establish scaling limits, proving convergence of origami maps to a maximal surface in the Minkowski space $\mathbb{R}^{2,1}$, and we verify structural rigidity conditions, leading to a new proof of convergence of height fluctuations to the Gaussian free field.

math.PR↗

arXiv2023

Perfect t-embeddings of uniformly weighted Aztec diamonds and tower graphs

In this work we study a sequence of perfect t-embeddings of uniformly weighted Aztec diamonds. We show that these perfect t-embeddings can be used to prove convergence of gradients of height fluctuations to those of the Gaussian free field. In particular we provide a first proof of the existence of a model satisfying all conditions of the main theorem of arXiv:2109.06272. This confirms the prediction of arXiv:2002.07540. An important part of our proof is to exhibit exact integral formulas for perfect t-embeddings of uniformly weighted Aztec diamonds. In addition, we construct and analyze perfect t-embeddings of another sequence of uniformly weighted finite graphs called tower graphs. Although we do not check all technical assumptions of the mentioned theorem for these graphs, we use perfect t-embeddings to derive a simple transformation which identifies height fluctuations on the tower graph with those of the Aztec diamond.

math-ph↗

arXiv2022

Dimer model and holomorphic functions on t-embeddings of planar graphs

We introduce the framework of discrete holomorphic functions on t-embeddings of weighted bipartite planar graphs; t-embeddings also appeared under the name Coulomb gauges in a recent paper arXiv:1810.05616. We argue that this framework is particularly relevant for the analysis of scaling limits of the height fluctuations in the corresponding dimer models. In particular, it unifies both Kenyon's interpretation of dimer observables as derivatives of harmonic functions on T-graphs and the notion of s-holomorphic functions originated in Smirnov's work on the critical Ising model. We develop an a priori regularity theory for such functions and provide a meta-theorem on convergence of the height fluctuations to the Gaussian Free Field. We also discuss how several more standard discretizations of complex analysis fit this general framework.

math.PR↗

arXiv2022

Lozenge tilings and the Gaussian free field on a cylinder

We use the periodic Schur process, introduced in arXiv:math/0601019v1, to study the random height function of lozenge tilings (equivalently, dimers) on an infinite cylinder distributed under two variants of the $q^{\operatorname{vol}}$ measure. Under the first variant, corresponding to random cylindric partitions, the height function converges to a deterministic limit shape and fluctuations around it are given by the Gaussian free field in the conformal structure predicted by the Kenyon-Okounkov conjecture. Under the second variant, corresponding to an unrestricted dimer model on the cylinder, the fluctuations are given by the same Gaussian free field with an additional discrete Gaussian shift component. Fluctuations of the latter type have been previously conjectured for dimer models on planar domains with holes.

math.PR↗

arXiv2021

Bipartite dimer model: perfect t-embeddings and Lorentz-minimal surfaces

This is the second paper in the series devoted to the study of the dimer model on t-embeddings of planar bipartite graphs. We introduce the notion of perfect t-embeddings and assume that the graphs of the associated origami maps converge to a Lorentz-minimal surface $\mathrm{S}_ξ$ as $δ\to 0$. In this setup we prove (under very mild technical assumptions) that the gradients of the height correlation functions converge to those of the Gaussian Free Field defined in the intrinsic metric of the surface $\mathrm{S}_ξ$. We also formulate several open questions motivated by our work.

math.PR↗

arXiv2020

Dimers and Circle patterns

We establish a correspondence between the dimer model on a bipartite graph and a circle pattern with the combinatorics of that graph, which holds for graphs that are either planar or embedded on the torus. The set of positive face weights on the graph gives a set of global coordinates on the space of circle patterns with embedded dual. Under this correspondence, which extends the previously known isoradial case, the urban renewal (local move for dimer models) is equivalent to the Miquel move (local move for circle patterns). As a consequence the Miquel dynamics on circle patterns is governed by the octahedron recurrence. As special cases of these circle pattern embeddings, we recover harmonic embeddings for resistor networks and s-embeddings for the Ising model.

math-ph↗

arXiv2019

Dominos in hedgehog domains

We introduce a new class of discrete approximations of planar domains that we call "hedgehog domains". In particular, this class of approximations contains two-step Aztec diamonds and similar shapes. We show that fluctuations of the height function of a random dimer tiling on hedgehog discretizations of a planar domain converge in the scaling limit to the Gaussian Free Field with Dirichlet boundary conditions. Interestingly enough, in this case the dimer model coupling function satisfies the same Riemann-type boundary conditions as fermionic observables in the Ising model. In addition, using the same factorization of the double-dimer model coupling function as in [17], we show that in the case of approximations by hedgehog domains the expectation of the double-dimer height function is harmonic in the scaling limit.

math-ph↗

arXiv2018

Dimers in piecewise Temperleyan domains

We study the large-scale behavior of the height function in the dimer model on the square lattice. Richard Kenyon has shown that the fluctuations of the height function on Temperleyan discretizations of a planar domain converge in the scaling limit (as the mesh size tends to zero) to the Gaussian Free Field with Dirichlet boundary conditions. We extend Kenyon's result to a more general class of discretizations. Moreover, we introduce a new factorization of the coupling function of the double-dimer model into two discrete holomorphic functions, which are similar to discrete fermions defined in [Stas, Stas07].For Temperleyan discretizations with appropriate boundary modifications, the results of Kenyon imply that the expectation of the double-dimer height function converges to a harmonic function in the scaling limit. We use the above factorization to extend this result to the class of all polygonal discretizations, that are not necessarily Temperleyan. Furthermore, we show that, quite surprisingly, the expectation of the double-dimer height function in the Temperleyan case is exactly discrete harmonic (for an appropriate choice of Laplacian) even before taking the scaling limit.

math-ph↗