Asymptotics of height change on toroidal Temperleyan dimer models

Abstract

The dimer model is an exactly solvable model of planar statistical mechanics. In its critical phase, various aspects of its scaling limit are known to be described by the Gaussian free field. For periodic graphs, criticality is an algebraic condition on the spectral curve of the model, determined by the edge weights; isoradial graphs provide another class of critical dimer models, in which the edge weights are determined by the local geometry. In the present article, we consider another class of graphs: general Temperleyan graphs, i.e. graphs arising in the (generalized) Temperley bijection between spanning trees and dimer models. Building in particular on Forman's formula and representations of Laplacian determinants in terms of Poisson operators, and under a minimal assumption - viz. that the underlying random walk converges to Brownian motion - we show that the natural topological observable on macroscopic tori converges in law to its universal limit, i.e. the law of the periods of the dimer height function converges to that of the periods of a compactified free field.

arXiv comment26 pages

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Cited by in corpus (6)

• Haldane relation for interacting dimers
• Asymptotics of the determinant of discrete Laplacians on triangulated and quadrangulated surfaces
• A characterisation of the Gaussian free field
• Lozenge tilings and the Gaussian free field on a cylinder
• Discrete and zeta-regularized determinants of the Laplacian on polygonal domains with Dirichlet boundary conditions
• The boundary disorder correlation for the Ising model on a cylinder