Berezin density and planar orthogonal polynomials

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.