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 , ⋅ ) B(z,\cdot ) for the polynomial Bergman space, where the point z ∈ C z\in \mathbb {C} is fixed. When z = ∞ z=\infty , the Berezin density is expressed in terms of the squared modulus of the corresponding normalized orthogonal polynomial P 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 |P|^2 in place of P ¯ \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 ) B(z,w) in the off-spectral regime, that is, when z 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.