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.Comment: 45 pages, 18 figures. v2: minor edits to introduction, fixed typo

Conformal invariance and universality of the dimer model

This thesis is dedicated to the study of the conformal invariance and the universality of the dimer model on planar bipartite graphs. Kenyon [41, 42] has established the conformal invariance of the limiting distribution of the dimer height function in the case of Temperleyan discretizations, discrete domains on the square lattice with special boundary conditions. In the thesis, we extended Kenyon’s result for more general classes of approximations on the square lattice. Yet another direction of research in the dimer model is the universality (which means that the scaling limit is independent of the shape of the lattice) of the planar dimer model. We describe how to construct a circle pattern embedding of a dimer planar graph using its Kasteleyn weights. We also introduce the definition of discrete holomorphicity on such an embedding. We focus on understanding the link between these functions and actual continuous holomorphic functions to study holomorphic observables of the dimer model

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 work 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

Mini-Workshop: Dimers, Ising and Spanning Trees beyond the Critical Isoradial Case

International audienc

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 work 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

Lozenge Tilings and the Gaussian Free Field on a Cylinder

Abstract We use the periodic Schur process, introduced in (Borodin in Duke Math J 140(3):391–468 2007), to study the random height function of lozenge tilings (equivalently, dimers) on an infinite cylinder distributed under two variants of the $$q^{{{\text {vol}}}}$$ q 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