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

Let $$\xi _0,\xi _1,\ldots $$ξ0,ξ1,… be i.i.d. random variables with zero mean and unit variance. We study the following four families of random analytic functions: $$\begin{aligned} P_n(z) := {\left\{ \begin{array}{ll} \sum \nolimits _{k=0}^n \sqrt{\left( {\begin{array}{c}n\\ k\end{array}}\right) } \xi _k z^k &{}\text { (spherical polynomials)},\\ \sum \nolimits _{k=0}^\infty \sqrt{\frac{n^k}{k!}} \xi _k z^k &{}\text { (flat random analytic function)},\\ \sum \nolimits _{k=0}^\infty \sqrt{\left( {\begin{array}{c}n+k-1\\ k\end{array}}\right) } \xi _k z^k &{}\text { (hyperbolic random analytic functions)},\\ \sum \nolimits _{k=0}^n \sqrt{\frac{n^k}{k!}} \xi _k z^k &{}\text { (Weyl polynomials)}. \end{array}\right. } \end{aligned}$$Pn(z):=∑k=0nnkξkzk(spherical polynomials),∑k=0∞nkk!ξkzk(flat random analytic function),∑k=0∞n+k-1kξkzk(hyperbolic random analytic functions),∑k=0nnkk!ξkzk(Weyl polynomials).We compute explicitly the limiting mean density of real zeroes of these random functions. More precisely, we provide a formula for $$\lim _{n\rightarrow \infty } n^{-1/2}\mathbb {E} N_n[a,b]$$limn→∞n-1/2ENn[a,b], where $$N_n[a,b]$$Nn[a,b] is the number of zeroes of $$P_n$$Pn in the interval [a, b].