Max von Horssen: Towards asymptotics of the correlation kernel for a family of 3×3-periodic hexagon tilings with merging gas phases

Abstract: We study a one-parameter family of tiling models of non-regular hexagons with doubly periodic weights, in which two side lengths are equal and their ratio to the third varies. These models exhibit three phases in the large-size limit: solid, liquid, and gas. Depending on the side length ratio, there are either one or two gas phases, and we observe that two gas phases can merge in two distinct ways.

We present an overview of a systematic method for obtaining large-size asymptotics of the correlation kernel and employ it to establish a scaling limit of the correlation kernel. The method includes non-linear steepest descent analysis of a Riemann-Hilbert problem and saddle point analysis on the (double) amoeba of a genus-one Harnack curve.

This talk is based on joint work with Arno Kuijlaars.