Zeros of Random Reinhardt Polynomials

Abstract

For a strictly pseudoconvex Reinhardt domain Ω with smooth boundary in C m+1 and a positive smooth measure µ on the boundary of Ω , we consider the ensemble PN of polynomials of degree N with the Gaussian probability measure γN which is induced by L 2 (∂Ω, dµ). Our aim is to compute the scaling limit distribution function and scaling limit pair correlation function for zeros near a point z ∈ ∂Ω. First, we apply the stationary phase method to the Boutet de Monvel-Sj¨ostrand Theorem to get the asymptotic for the scaling limit partial Szeg¨o kernel around z ∈ ∂Ω. Then by using the Kac-Rice formula, we compute the scaling limit distribution and pair correlation functions.