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

Research paper by **Hendrik Flasche, Zakhar Kabluchko**

Indexed on: **07 Feb '18**Published on: **07 Feb '18**Published in: **arXiv - Mathematics - Probability**

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