Random planar dimer models (or equivalently tiling models) have been a subject of extensive research in mathematics and physics for several decades. Recent progress has been made in understanding the behavior of planar dimer models with periodic edge weights, particularly models on the subgraph of the square lattice known as the Aztec diamond graph. In this talk, I will discuss the effect of a temperature parameter in the doubly periodic Aztec diamond dimer model in the zero temperature limit. In this limit, the Aztec diamond undergoes crystallization: The limit shape converges to a piecewise linear function we call the tropical limit shape, and the local fluctuations are governed by the Gibbs measures with the slope dictated by the tropical limit shape for low enough temperature.

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 to Kirchhoff's problem on the tropical curve.

Based on joint work with Alexei Borodin.