Matrix-valued orthogonal polynomials related to hexagon tilings

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 ${\tilde{\Sigma }}_0$ 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 ${\tilde{\Sigma }}_0$ that we were unable to prove, but that is convincingly supported by numerical evidence.

31 pages, 10 figures

References in corpus (5)

• Normal random matrix ensemble as a growth problem
• Viscous shocks in Hele-Shaw flow and Stokes phenomena of the Painleve I transcendent
• Shocks and finite-time singularities in Hele-Shaw flow
• An equilibrium problem on the sphere with two equal charges
• Logarithmic Equilibrium on the Sphere in the Presence of Multiple Point Charges

Cited by in corpus (3)

• Partition functions of determinantal and Pfaffian Coulomb gases with radially symmetric potentials
• Spherical Induced Ensembles with Symplectic Symmetry
• Orthogonal polynomials in the normal matrix model with two insertions