Matrix-valued orthogonal polynomials related to hexagon tilings

TL;DR

This paper explores matrix-valued orthogonal polynomials linked to hexagon tilings, revealing their asymptotic zero distributions and phase transition behaviors through connections with scalar orthogonal polynomials.

Contribution

It introduces a novel analysis of MVOPs related to hexagon tilings, especially in 2-periodic cases, and characterizes their asymptotic zero distributions and phase transition phenomena.

Findings

Zeros tend to a self-intersecting curve in the complex plane.

Asymptotic behavior of MVOPs reflects phase transitions in the tiling model.

Numerical evidence supports the conjectured zero distribution of matrix entries.

Abstract

In this paper, we study a class of matrix-valued orthogonal polynomials (MVOPs) that are related to 2-periodic lozenge tilings of a hexagon. The general model depends on many parameters. In the cases of constant and $2$-periodic parameter values we show that the MVOP can be expressed in terms of scalar polynomials with non-Hermitian orthogonality on a closed contour in the complex plane. The 2-periodic hexagon tiling model with a constant parameter has a phase transition in the large size limit. This is reflected in the asymptotic behavior of the MVOP as the degree tends to infinity. The connection with the scalar orthogonal polynomials allows us to find the limiting behavior of the zeros of the determinant of the MVOP. The zeros tend to a curve $\widetilde{\Sigma}_0$ in the complex plane that has a self-intersection. The zeros of the individual entries of the MVOP show a different behavior and we find the limiting zero distribution of the upper right entry under a geometric condition on the curve $\widetilde{\Sigma}_0$ that we were unable to prove, but that is convincingly supported by numerical evidence.