Lozenge tilings and Hurwitz numbers

TL;DR

This paper presents a novel proof that the distribution of vertical tiles near a turning point in large lozenge tilings matches GUE eigenvalues, using combinatorial methods and Hurwitz numbers instead of traditional integrable probability tools.

Contribution

It introduces a new combinatorial approach to connect lozenge tilings with GUE eigenvalue distributions via Hurwitz numbers, bypassing standard integrable probability techniques.

Findings

Vertical tiles near a turning point follow GUE eigenvalue distribution.

The proof employs a combinatorial interpretation of the Harish-Chandra/Itzykson-Zuber integral.

Establishes a new link between tiling models and random matrix theory.

Abstract

We give a new proof of the fact that, near a turning point of the frozen boundary, the vertical tiles in a uniformly random lozenge tiling of a large sawtooth domain are distributed like the eigenvalues of a GUE random matrix. Our argument uses none of the standard tools of integrable probability. In their place, it uses a combinatorial interpretation of the Harish-Chandra/Itzykson-Zuber integral as a generating function for desymmetrized Hurwitz numbers.