A Universality Property of Gaussian Analytic Functions

Abstract

We consider random analytic functions defined on the unit disk of the complex plane \(f(z) = \sum_{n=0}^{\infty} a_{n} X_{n} z^{n}\), where the Xn’s are i.i.d., complex-valued random variables with mean zero and unit variance. The coefficients an are chosen so that f(z) is defined on a domain of ℂ carrying a planar or hyperbolic geometry, and \(\mathbf{E}f(z)\overline{f(w)}\) is covariant with respect to the isometry group. The corresponding Gaussian analytic functions have been much studied, and their zero sets have been considered in detail in a monograph by Hough, Krishnapur, Peres, and Virag. We show that for non-Gaussian coefficients, the zero set converges in distribution to that of the Gaussian analytic functions as one transports isometrically to the boundary of the domain. The proof is elementary and general.