Asymptotics of height change on toroidal Temperleyan dimer models

TL;DR

This paper investigates the asymptotic behavior of height changes in Temperleyan dimer models on toroidal graphs, showing convergence to a universal limit described by a free field under minimal assumptions.

Contribution

It extends the understanding of dimer models by analyzing height change asymptotics on general Temperleyan graphs using Laplacian determinants and Brownian motion convergence.

Findings

Height change distributions converge to a universal free field limit.

Results apply to a broad class of Temperleyan graphs beyond isoradial cases.

The convergence holds under minimal assumptions about the underlying random walk.

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.