On the asymptotics of dimers on tori

TL;DR

This paper investigates the asymptotic behavior of the dimer model on large toric graphs, revealing universal finite-size corrections and new insights into loop distributions in double-dimer models.

Contribution

It provides the first detailed asymptotic expansion of the dimer partition function on large tori, including finite-size corrections depending on conformal shape and parity.

Findings

Asymptotic expansion of the dimer partition function with finite-size corrections

Finite-size correction depends on conformal shape and parity information

New insights into loop winding distributions in double-dimer models

Abstract

We study asymptotics of the dimer model on large toric graphs. Let $\mathbb L$ be a weighted $\mathbb{Z}^2$-periodic planar graph, and let $\mathbb{Z}^2 E$ be a large-index sublattice of $\mathbb{Z}^2$. For $\mathbb L$ bipartite we show that the dimer partition function on the quotient $\mathbb{L}/(\mathbb{Z}^2 E)$ has the asymptotic expansion $\exp[A f_0 + \text{fsc} + o(1)]$, where $A$ is the area of $\mathbb{L}/(\mathbb{Z}^2 E)$, $f_0$ is the free energy density in the bulk, and $\text{fsc}$ is a finite-size correction term depending only on the conformal shape of the domain together with some parity-type information. Assuming a conjectural condition on the zero locus of the dimer characteristic polynomial, we show that an analogous expansion holds for $\mathbb{L}$ non-bipartite. The functional form of the finite-size correction differs between the two classes, but is universal within each class. Our calculations yield new information concerning the distribution of the number of loops winding around the torus in the associated double-dimer models.