Various convergence results for the Bergman kernel of the Hilbert space of
all polynomials in \C^{n} of total degree at most k, equipped with a weighted
norm, are obtained. The weight function is assumed to be C^{1,1}, i.e. it is
differentiable and all of its first partial derivatives are locally Lipshitz
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-Ampere measure of the weight function itself on a
certain set. A setting of polynomials associated to a given Newton polytope,
scaled by k, is also considered. These results apply directly to the study of
the distribution of zeroes of random polynomials and of the eigenvalues of
random normal matrices.Comment: v1: 11 pages v2: 19 pages. Substantial revision: regularity
assumption on the weight weakened to C^1,1, setting of polynomials with a
given Newton polytope considered, examples and a figure adde
