Matrix valued orthogonal polynomials arising from hexagon tilings with 3x3-periodic weightings

TL;DR

This paper studies matrix valued orthogonal polynomials from a special class of hexagon tilings with 3x3 periodic weights, analyzing their asymptotic behavior and zero distribution using Riemann-Hilbert techniques.

Contribution

It introduces a new class of MVOP linked to 3x3 periodic hexagon tilings and derives their asymptotics through explicit equilibrium measures and steepest descent analysis.

Findings

Explicit equilibrium measure for the tilings

Asymptotic zero distribution of MVOP

Strong asymptotic formulas derived from Riemann-Hilbert analysis

Abstract

Matrix valued orthogonal polynomials (MVOP) appear in the study of doubly periodic tiling models. Of particular interest is their limiting behavior as the degree tends to infinity. In recent years, MVOP associated with doubly periodic domino tilings of the Aztec diamond have been successfully analyzed. The MVOP related to doubly periodic lozenge tilings of a hexagon are more complicated. In this paper we focus on a special subclass of hexagon tilings with 3x3 periodicity. The special subclass leads to a genus one spectral curve with additional symmetries that allow us to find an equilibrium measure in an external field explicitly. The equilibrium measure gives the asymptotic distribution for the zeros of the determinant of the MVOP. The associated g-functions appear in the strong asymptotic formula for the MVOP that we obtain from a steepest descent analysis of the Riemann-Hilbert problem for MVOP.