Airy process at a thin rough region between frozen and smooth

TL;DR

This paper demonstrates that at the boundary between rough and smooth regions of a two-periodic Aztec diamond, the last path converges to the Airy process under specific scaling conditions, revealing new insights into the boundary behavior.

Contribution

It establishes the convergence of the last path to the Airy process at a thin rough-smooth boundary when the parameter tends to zero at a certain rate.

Findings

Last path converges to the Airy process at the boundary.

Rough region has a mesoscopic width under the scaling.

Dimer correlations are described by a discrete Bessel kernel at microscopic width.

Abstract

We show there is a last path at the rough smooth boundary of the two-periodic Aztec diamond with parameter $a\in (0,1)$ that, suitably rescaled, converges to the Airy process, under the condition that $a$ tends to zero as the size of the Aztec diamond tends to infinity at a certain rate. This condition causes the rough region to have a thin, mesoscopic width. We also show that the dimers are described by a discrete Bessel kernel when the width is only of microscopic size.