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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 [ab].

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Fig. 1

Notes

  1. 1.

    A random variable Z has negative binomial distribution \(\text {NBin}(n,p)\) if \(\mathbb {P}\left[ Z=k\right] = \left( {\begin{array}{c}n+k-1\\ k\end{array}}\right) p^n(1-p)^k\) for \(k=0,1,\ldots \).

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Acknowledgements

The support by the SFB 878 “Groups, Geometry and Actions” is gratefully acknowledged.

Author information

Correspondence to Zakhar Kabluchko.

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Cite this article

Flasche, H., Kabluchko, Z. Real Zeroes of Random Analytic Functions Associated with Geometries of Constant Curvature. J Theor Probab 33, 103–133 (2020). https://doi.org/10.1007/s10959-018-0843-z

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Keywords

  • Random polynomials
  • Random analytic functions
  • Spherical polynomials
  • Flat analytic function
  • Hyperbolic analytic function
  • Weyl polynomials
  • Real zeroes
  • Weak convergence
  • Gaussian processes
  • Functional limit theorem

Mathematics Subject Classification (2010)

  • Primary: 30C15
  • 26C10
  • Secondary: 60F99
  • 60F17
  • 60F05
  • 60G15