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

TL;DR

This paper derives explicit formulas for the asymptotic mean density of real zeros of four families of random analytic functions associated with geometries of constant curvature, as the degree tends to infinity.

Contribution

It provides the first explicit computation of the limiting mean density of real zeros for these four classes of random analytic functions.

Findings

Explicit formulas for the limiting mean density of real zeros.

Asymptotic behavior of zero counts scaled by n^{-1/2}.

Connections to 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 $[a,b]$.