Real zeros of random analytic functions associated with geometries of constant curvature

Abstract

Let $\xi_0, \xi_1, \dots$ be i.i.d. random variables with zero mean and unit variance. We study the following four families of random analytic functions: $\sum_{k=0}^n \sqrt{\binom nk} \xi_k z^k$ (spherical polynomials), $\sum_{k=0}^\infty \sqrt{\frac{n^k}{k!}} \xi_k z^k$ (flat random analytic function), $\sum_{k=0}^\infty \sqrt{\binom {n+k-1} k} \xi_k z^k$ (hyperbolic random analytic functions), $\sum_{k=0}^n \sqrt{\frac{n^k}{k!}} \xi_k z^k$ (Weyl polynomials). We compute explicitly the limiting mean density of real zeroes of these random functions. More precisely, we provide a formula for $\lim_{n\to\infty} n^{-1/2} \mathbb{E}N_n[a,b]$, where $N_n[a, b]$ is the number of zeroes in the interval $[a,b]$.