Local Statistics and Shuffling for Dimers on a Square-Hexagon Lattice

TL;DR

This paper analyzes the local statistics of dimers on tower graphs derived from the square-hexagon lattice, confirming their convergence to translation-invariant Gibbs measures and connecting the model to the Anisotropic KPZ universality class.

Contribution

It proves the convergence of local statistics to Gibbs measures and introduces a growth process linked to the dimer model on tower graphs.

Findings

Local statistics converge to translation-invariant Gibbs measures.

The growth process's distribution matches the dimer model on tower graphs.

The model belongs to the Anisotropic KPZ universality class.

Abstract

We study the dimer model on special subgraphs of the square hexagon lattice called "tower graphs" of size $N$. Using integrable probability techniques, we confirm that as $N \rightarrow \infty$, the local statistics are translation invariant Gibbs measures, as conjectured by Kenyon-Okounkov-Sheffield. We also present a 2+1-dimensional discrete time growth process, whose time $N$ distribution is exactly the dimer model on the size $N$ tower, and we compute the current of this growth process and confirm that the model belongs to the Anisotropic KPZ universality class.