Large deviations of empirical zero point measures on Riemann surfaces, I: $g = 0$

Abstract

We prove an LDP for the empirical measure of complex zeros of a Gaussian random complex polynomial of degree N of one variable as N tends to infinity. The Gaussian measure is induced by an inner product defined by a smooth weight (Hermitian metric) $h$ and a Bernstein-Markov measure $\nu$. The speed is N^2 and the the unique minimizer of the rate function $I$ is the weighted equilibrium measure $\nu_{h, K}$ with respect to $h$ on the support $K$ of $\nu$.