A universality property of Gaussian analytic functions
This paper deals with random analytic functions (RAF), a class of functions which contains also the well-studied Gaussian analytic functions (GAF) (see, e.g., [\textit{J. B. Hough} et al., Zeros of Gaussian analytic functions and determinantal point processes. Providence, RI: American Mathematical Society (AMS) (2008; Zbl 1190.60038)]). In the first section, the authors introduce their framework, they discuss some known results about these two families of functions and they state a theorem for the zero set of GAF due to \textit{Y. Peres} and \textit{B. Virág} [Acta Math. 194, No. 1, 1--35 (2005; Zbl 1099.60037)]. They also announce their main result concerning convergence in distribution of the zero set of non-Gaussian RAF to that of GAF when we move isometrically to the boundary of the domain. Section 2 is devoted to the proof of the main theorem by using the CLT and classical results in complex analysis. Finally, a simple application for a discrete family of RAF is provided in Section 3, including also the plots of two particular examples.
scientific article; zbMATH DE number 6050429
scientific article