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

Research paper by Hendrik Flasche, Zakhar Kabluchko

Indexed on: 18 Jul '18Published on: 14 Jul '18Published in: Journal of Theoretical Probability



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].