Airy point process at the liquid-gas boundary

TL;DR

This paper demonstrates that the liquid-gas boundary in a two-periodic Aztec diamond tiling converges to the extended Airy point process, suggesting a universal description similar to the liquid-solid boundary.

Contribution

It introduces a new random measure to analyze the liquid-gas boundary and proves its convergence to the extended Airy point process, revealing a universal behavior.

Findings

The random measure converges to the extended Airy point process.

The liquid-gas boundary exhibits long-range correlations.

The Airy process describes the liquid-gas boundary in this model.

Abstract

Domino tilings of the two-periodic Aztec diamond feature all of the three possible types of phases of random tiling models. These phases are determined by the decay of correlations between dominoes and are generally known as solid, liquid and gas. The liquid-solid boundary is easy to define microscopically and is known in many models to be described by the Airy process in the limit of a large random tiling. The liquid-gas boundary has no obvious microscopic description. Using the height function we define a random measure in the two-periodic Aztec diamond designed to detect the long range correlations visible at the liquid-gas boundary. We prove that this random measure converges to the extended Airy point process. This indicates that, in a sense, the liquid-gas boundary should also be described by the Airy process.