Crystallization of the Aztec diamond

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.