The Airy line ensemble at the rough-smooth boundary

TL;DR

This paper investigates the rough-smooth boundary in a two-periodic Aztec diamond tiling, showing convergence of the height function to an Airy surface plus noise, and proving paths converge to the Airy line ensemble.

Contribution

It provides the first convergence result for undirected paths to the Airy line ensemble and analyzes the boundary fluctuations in a domino tiling model.

Findings

Height function converges to an Airy surface plus noise.

Temperleyan backbone paths converge to the Airy line ensemble.

First convergence result for undirected paths to the Airy line ensemble.

Abstract

We study the rough-smooth boundary in the two-periodic Aztec diamond, a random domino tiling model exhibiting three types of macroscopic regions. We show that the height function at this boundary converges to an independent sum of an Airy surface and an i.i.d. noise field with fluctuations governed by the full-plane smooth phase. Going further, we prove convergence of a family of Temperleyan backbone paths to the Airy line ensemble. This gives the first convergence result for a family of undirected paths converging to the Airy line ensemble, as well as Airy convergence at a noisy boundary.