Third-order phase transition in random tilings

TL;DR

This paper demonstrates a third-order phase transition in the free energy of domino tilings of an Aztec diamond with a cut-off corner, occurring at the arctic ellipse, analyzed through correlation functions and matrix models.

Contribution

It reveals a third-order phase transition in domino tilings with a cut-off corner, linking tiling properties to nonlocal correlation functions and matrix models.

Findings

Free energy exhibits a third-order phase transition at the arctic ellipse.

The transition shares properties with Gross-Witten-Wadia and Douglas-Kazakov phase transitions.

Analysis uses tau-functions and random matrix model integrals for correlation functions.

Abstract

We consider the domino tilings of an Aztec diamond with a cut-off corner of macroscopic square shape and given size, and address the bulk properties of tilings as the size is varied. We observe that the free energy exhibits a third-order phase transition when the cut-off square, increasing in size, reaches the arctic ellipse---the phase separation curve of the original (unmodified) Aztec diamond. We obtain this result by studying the thermodynamic limit of certain nonlocal correlation function of the underlying six-vertex model with domain wall boundary conditions, the so-called emptiness formation probability (EFP). We consider EFP in two different representations: as a tau-function for Toda chains and as a random matrix model integral. The latter has a discrete measure and a linear potential with hard walls; the observed phase transition shares properties with both Gross-Witten-Wadia and Douglas-Kazakov phase transitions.