Fock's dimer model on the Aztec diamond

TL;DR

This paper derives explicit formulas for the inverse Kasteleyn matrix and partition function of Fock's dimer model on the Aztec diamond, extending limit shape results and connecting them to spectral curve amoebas.

Contribution

It provides explicit formulas for the inverse Kasteleyn matrix and partition function for Fock's dimer model on the Aztec diamond, extending previous results to non-generic weights.

Findings

Explicit inverse Kasteleyn matrix formula derived

Partition function formula obtained, recovering Stanley's formula

Extended limit shape results to non-generic weights

Abstract

We consider the dimer model on the Aztec diamond with Fock's weights, which is gauge equivalent to the model with any choice of positive weight function. We prove an explicit, compact formula for the inverse Kasteleyn matrix, thus extending numerous results in the case of periodic graphs. We also show an explicit product formula for the partition function; as a specific instance of the genus 0 case, we recover Stanley's formula. We then use our explicit formula for the inverse Kasteleyn matrix to recover, in a simple way, limit shape results; we also obtain new ones. In doing so, we extend the correspondence between the limit shape and the amoeba of the corresponding spectral curve of arXiv:2306.07482 to the case of non-generic weights.