Berezin density and planar orthogonal polynomials

TL;DR

This paper develops a nonlinear potential theory approach to analyze Berezin densities and orthogonal polynomials in polynomial Bergman spaces, extending previous methods to asymptotic regimes and off-spectral analysis.

Contribution

It introduces a nonlinear potential problem characterizing Berezin densities and adapts the soft Riemann-Hilbert method to study asymptotics of orthogonal polynomials with exponential weights.

Findings

Characterization of Berezin density via nonlinear potential theory

Asymptotic analysis of orthogonal polynomials with exponential weights

Extension of potential theory methods to off-spectral regimes

Abstract

We introduce a nonlinear potential theory problem for the Laplacian, the solution of which characterizes the Berezin density $B(z,\cdot)$ for the polynomial Bergman space, where the point $z\in\mathbb{C}$ is fixed. When $z=\infty$, the Berezin density is expressed in terms of the squared modulus of the corresponding normalized orthogonal polynomial $P$. We use an approximate version of this characterization to study the asymptotics of the orthogonal polynomials in the context of exponentially varying weights. This builds on earlier works by Its-Takhtajan and by the first author on a soft Riemann-Hilbert problem for planar orthogonal polynomials, where in place of the Laplacian we have the $\bar\partial$-operator. We adapt the soft Riemann-Hilbert approach to the nonlinear potential problem, where the nonlinearity is due to the appearance of $|P|^2$ in place of $\overline{P}$. Moreover, we suggest how to adapt the potential theory method to the study of the asymptotics of more general Berezin densities $B(z,w)$ in the off-spectral regime, that is, when $z$ is fixed outside the droplet. This is a first installment in a program to obtain an explicit global expansion formula for the polynomial Bergman kernel, and, in particular, of the one-point function of the associated random normal matrix ensemble.