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

Hendrik Flasche, Zakhar Kabluchko

Published 2018-02-07Version 1

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: $$ P_n(z) := \begin{cases} \sum_{k=0}^n \sqrt{\binom nk} \xi_k z^k &\text{ (spherical polynomials)}, \sum_{k=0}^\infty \sqrt{\frac{n^k}{k!}} \xi_k z^k &\text{ (flat random analytic function)}, \sum_{k=0}^\infty \sqrt{\binom {n+k-1} k} \xi_k z^k &\text{ (hyperbolic random analytic functions)}, \sum_{k=0}^n \sqrt{\frac{n^k}{k!}} \xi_k z^k &\text{ (Weyl polynomials)}. \end{cases} $$ 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 of $P_n$ in the interval $[a,b]$.