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: ${\sum }_{k=0}^n\sqrt{\left(\genfrac{}{}{0ex}{}{n}{k}\right)}{\xi }_k{z}^k$ (spherical polynomials), ${\sum }_{k=0}^{\infty }\sqrt{\frac{n^k}{k!}}{\xi }_k{z}^k$ (flat random analytic function), ${\sum }_{k=0}^{\infty }\sqrt{\left(\genfrac{}{}{0ex}{}{n+k-1}{k}\right)}{\xi }_k{z}^k$ (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 $[a,b]$.

26 pages, 1 figure