Simple matrix models for random Bergman metrics

Abstract

Recently, the authors have proposed a new approach to the theory of random metrics,making an explicit link between probability measures on the space of metrics ona Kähler manifold and random matrix models. We consider simple examples ofsuch models and study the one- and two-point functions of the metric. Thesegeometric correlation functions correspond to new interesting types of matrix modelcorrelators. We provide in particular a detailed study of the Wishart model, where wedetermine the correlation functions explicitly. We find that the random measure inthis model turns out to be concentrated on the background metric in the largeN limit.