IUMJ

Title: Bergman kernels for weighted polynomials and weighted equilibrium measures of $\mathbb{C}^{n}$

Authors: Robert J. Berman

Issue: Volume 58 (2009), Issue 4, 1921-1946

Abstract:

Various convergence results for the Bergman kernel of the Hilbert space of all polynomials in $\mathbb{C}^{n}$ of total degree at most $k$, equipped with a weighted norm, are obtained. The weight function $\phi$ is assumed to be $\mathcal{C}^{1,1}$, i.e. $\phi$ is differentiable and all of its first partial derivatives are locally Lipschitz continuous. The convergence is studied in the large $k$ limit and is expressed in terms of the global equilibrium potential associated to the weight function, as well as in terms of the Monge-Amp\'ere measure of the weight function itself on a certain set. These results apply directly to the study of the distribution of zeroes of random polynomials and of the eigenvalues of random normal matrices.