Critical measures on higher genus Riemann surfaces

TL;DR

This paper extends the theory of critical measures from the complex plane to higher genus Riemann surfaces, analyzing their structure via bipolar Green's energy and applications to random tilings.

Contribution

It develops a new framework for critical measures on higher genus Riemann surfaces using bipolar Green's energy and solves a max-min problem under certain conditions.

Findings

Existence of solutions to the max-min problem for bipolar Green's energy.

Critical measures supported on maximal trajectories of quadratic differentials in genus one.

Application to asymptotic analysis of matrix orthogonal polynomials in random tiling models.

Abstract

Critical measures in the complex plane are saddle points for the logarithmic energy with external field. Their local and global structure was described by Martinez-Finkelshtein and Rakhmanov. In this paper we start the development of a theory of critical measures on higher genus Riemann surfaces, where the logarithmic energy is replaced by the energy with respect to a bipolar Green's kernel. We study a max-min problem for the bipolar Green's energy with external fields Re V where dV is a meromorphic differential. Under reasonable assumptions the max-min problem has a solution and we show that the corresponding equilibrium measure is a critical measure in the external field. In a special genus one situation we are able to show that the critical measure is supported on maximal trajectories of a meromorphic quadratic differential. We are motivated by applications to random lozenge tilings of a hexagon with periodic weightings. Correlations in these models are expressible in terms of matrix valued orthogonal polynomials. The matrix orthogonality is interpreted as (partial) scalar orthogonality on a Riemann surface. The theory of critical measures will be useful for the asymptotic analysis of a corresponding Riemann-Hilbert problem as we outline in the paper.