Limit shapes from harmonicity: dominos and the five vertex model

TL;DR

This paper develops a method to construct limit shapes for domino tilings and the 5-vertex model using harmonic extension techniques, providing insights into their geometric and probabilistic structures.

Contribution

It introduces a harmonic extension approach to derive limit shapes for domino and 5-vertex models in polygonal domains, advancing the understanding of their asymptotic behavior.

Findings

Successful construction of limit shapes for domino tilings.

Application of harmonic extension method to the 5-vertex model.

Enhanced understanding of geometric structures in tiling models.

Abstract

We discuss how to construct limit shapes for the domino tiling model (square lattice dimer model) and $5$-vertex model, in appropriate polygonal domains. Our methods are based on the harmonic extension method of [R. Kenyon and I. Prause, Gradient variational problems in $\mathbb{R}^2$, Duke Math J. 2022].