Real Zeroes of Random Analytic Functions Associated with Geometries of Constant Curvature

Abstract

Let $\xi _0,\xi _1,\ldots$ be i.i.d. random variables with zero mean and unit variance. We study the following four families of random analytic functions:

$$\begin{aligned} P_n(z) := {\left\{ \begin{array}{ll} \sum \nolimits _{k=0}^n \sqrt{\left( {\begin{array}{c}n\\ k\end{array}}\right) } \xi _k z^k &{}\text { (spherical polynomials)},\\ \sum \nolimits _{k=0}^\infty \sqrt{\frac{n^k}{k!}} \xi _k z^k &{}\text { (flat random analytic function)},\\ \sum \nolimits _{k=0}^\infty \sqrt{\left( {\begin{array}{c}n+k-1\\ k\end{array}}\right) } \xi _k z^k &{}\text { (hyperbolic random analytic functions)},\\ \sum \nolimits _{k=0}^n \sqrt{\frac{n^k}{k!}} \xi _k z^k &{}\text { (Weyl polynomials)}. \end{array}\right. } \end{aligned}$$

We compute explicitly the limiting mean density of real zeroes of these random functions. More precisely, we provide a formula for $\lim _{n\rightarrow \infty } n^{-1/2}\mathbb {E} N_n[a,b]$, where $N_n[a,b]$ is the number of zeroes of $P_n$ in the interval [a, b].