Dimers in piecewise Temperleyan domains

TL;DR

This paper extends Kenyon's results on the convergence of the dimer height function to the Gaussian Free Field to more general discretizations and introduces a new factorization of the double-dimer model's coupling function, revealing exact harmonicity in certain cases.

Contribution

It generalizes the convergence results to polygonal discretizations and introduces a novel factorization of the double-dimer coupling function into discrete holomorphic functions.

Findings

Height function fluctuations converge to Gaussian Free Field in generalized discretizations.

New factorization of the double-dimer coupling function into discrete holomorphic functions.

Expectation of the double-dimer height function is exactly discrete harmonic in the Temperleyan case.

Abstract

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.