Crystallization of the Aztec diamond

TL;DR

This paper studies dimer models on Aztec diamonds with weighted edges, proving that at zero temperature they crystallize into a piecewise linear shape governed by tropical geometry, revealing new connections between statistical mechanics and tropical curves.

Contribution

It introduces the concept of tropical limit shapes and arctic curves for dimer models at zero temperature, linking them to tropical geometry and spectral curves.

Findings

Limit shape converges to a tropical piecewise linear function.

Local fluctuations are governed by Gibbs measures with specific slopes.

Tropical curves and action functions describe the zero-temperature limit geometry.

Abstract

We consider dimer models on growing Aztec diamonds, which are certain domains in the square lattice, with edge weights of the form $\nu(\,\cdot\,)^\beta$, where $\nu(\,\cdot\,)$ is a doubly periodic function on the edges of the lattice and $\beta$ is an inverse temperature parameter. We prove that in the zero-temperature ($\beta\to\infty$) limit, and for generic values of $\nu(\,\cdot\,)$, these dimer models undergo crystallization: The limit shape converges to a piecewise linear function called the tropical limit shape, and the local fluctuations are governed by the Gibbs measures with the slope dictated by the tropical limit shape for high enough values of $\beta$. We also show that the tropical limit shape and the tropical arctic curve (consisting of ridges of the crystal) are described in terms of a tropical curve and a tropical action function on that curve, which are the tropical analogs of the spectral curve and the action function that describe the finite-temperature models. The tropical curve is explicit in terms of the edge weights, and the tropical action function is a solution of Kirchhoff's problem on the tropical curve.