# A periodic hexagon tiling model and non-Hermitian orthogonal polynomials

## TL;DR

This paper analyzes a family of lozenge tiling models on hexagons, revealing phase transitions and disordered regions through asymptotic analysis of non-Hermitian orthogonal polynomials and Riemann-Hilbert techniques.

## Contribution

It introduces a detailed asymptotic analysis of lozenge tilings using non-Hermitian orthogonal polynomials, uncovering phase transition phenomena.

## Key findings

- Disordered regions are two disjoint ellipses at low temperature.
- At high temperature, the ellipses merge into a single region.
- A tacnode appears at the phase transition point.

## Abstract

We study a one-parameter family of probability measures on lozenge tilings of large regular hexagons that interpolates between the uniform measure on all possible tilings and a particular fully frozen tiling. The description of the asymptotic behavior can be separated into two regimes: the low and the high temperature regime. Our main results are the computations of the disordered regions in both regimes and the limiting densities of the different lozenges there. For low temperatures, the disorded region consists of two disjoint ellipses. In the high temperature regime the two ellipses merge into a single simply connected region. At the transition from the low to the high temperature a tacnode appears. The key to our asymptotic study is a recent approach introduced by Duits and Kuijlaars providing a double integral representation for the correlation kernel. One of the factors in the integrand is the Christoffel-Darboux kernel associated to polynomials that satisfy non-Hermitian orthogonality relations with respect to a complex-valued weight on a contour in the complex plane. We compute the asymptotic behavior of these orthogonal polynomials and the Christoffel-Darboux kernel by means of a Riemann-Hilbert analysis. After substituting the resulting asymptotic formulas into the double integral we prove our main results by classical steepest descent arguments.