Large deviations of empirical measures of zeros on Riemann surfaces

Abstract

This is a continuation of a project on large deviations for the empirical measures of zeros of random holomorphic sections of random line bundles over a Riemann surface X. In a previous article with O. Zeitouni (arXiv:0904.4271), we proved an LDP for random polynomials in the genus zero case. In higher genus, there is a Picard variety of line bundles and so the line bundle L is a random variable as well as the section s. The space of pairs (L, s) is known as the "vortex moduli space". The zeros of (L, s) fill out the configuration space $X^{(N)}$ of $N$ points of $X$. The LDP shows that the configurations concentrate at one equilibrium measure exponentially fast. The new features of the proof involve Abel-Jacobi theory, the prime form and bosonization.