SU(1,1) Random Polynomials

Abstract

We study statistical properties of zeros of random polynomials and random analytic functions associated with the pseudoeuclidean group of symmetries SU(1,1), by utilizing both analytical and numerical techniques. We first show that zeros of the SU(1,1) random polynomial of degree $N$ are concentrated in a narrow annulus of the order of $N^{-1}$ around the unit circle on the complex plane, and we find an explicit formula for the scaled density of the zeros distribution along the radius in the limit $N\to\infty$. Our results are supported through various numerical simulations. We then extend results of Hannay \cite{H} and Bleher, Shiffman, Zelditch \cite{BSZ2} to derive different formulae for correlations between zeros of the SU(1,1) random analytic functions, by applying the generalized Kac-Rice formula. We express the correlation functions in terms of some Gaussian integrals, which can be evaluated combinatorially as a finite sum over Feynman diagrams or as a supersymmetric integral. Due to the SU(1,1) symmetry, the correlation functions depend only on the hyperbolic distances between the points on the unit disk, and we obtain an explicit formula for the two point correlation function. It displays quadratic repulsion at small distances and fast decay of correlations at infinity. In an appendix to the paper we evaluate correlations between the outer zeros $|z_j|>1$ of the SU(1,1) random polynomial, and we prove that the inner and outer zeros are independent in the limit when the degree of the polynomial goes to infinity.