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 $\nu$.

References in corpus (3)

• Bergman kernels for weighted polynomials and weighted equilibrium measures of C^n
• Number variance of random zeros
• Growth of balls of holomorphic sections and energy at equilibrium

Cited by in corpus (6)

• Analytic torsion, vortices and positive Ricci curvature
• On the geometry of random lemniscates
• Stability and integration over Bergman metrics
• Large deviations for zeros of $P(\varphi {)}_2$ random polynomials
• Fekete points and convergence towards equilibrium measures on complex manifolds
• Random zeros on complex manifolds: conditional expectations