Limit shapes and harmonic tricks

TL;DR

This paper explains the tangent plane method for analyzing arctic curves in the dimer model and extends it to multiply connected domains, providing explicit elliptic function parametrizations of new arctic curves.

Contribution

It extends the tangent plane method to multiply connected regions and derives explicit elliptic function parametrizations of arctic curves with a hole.

Findings

Explicit parametrization of arctic curves in multiply-connected regions.

Visualization of the limit height functions.

First explicit elliptic function parametrization for these curves.

Abstract

This article has two main goals. First, it provides a self-contained exposition of the tangent plane method for the dimer model - a technique for analyzing arctic curves and limit shapes introduced by R. Kenyon and I. Prause (2020). Second, it extends this method to multiply connected domains through a nontrivial computation of the frozen boundary for the Aztec diamond with a hole. This computation yields the first explicit parametrization in terms of elliptic functions of a family of arctic curves of a multiply-connected region indexed by the height change (hole height). We also derive and visualize the corresponding limit height functions.