Distribution of Normalized Zero-Sets of Random Entire Functions

TL;DR

This paper investigates the distribution of normalized zero-sets of random entire functions, extending existing theories from polynomials to entire functions and drawing analogies with Nevanlinna's First Main Theorem.

Contribution

It generalizes the distribution results of zero-sets from polynomial cases to entire functions, connecting Nevanlinna theory with random holomorphic functions.

Findings

Generalization of zero-set distribution to entire functions

Extension of Shiffman and Zelditch's theory

Analogy with Nevanlinna's First Main Theorem

Abstract

This paper is concerned with the distribution of normalized zero-sets of random entire functions. The normalization of the zero-set is performed in the same way as that of the counting function for an entire function in Nevanlinna theory. The result generalizes the Shiffman and Zelditch theory on the distribution of the zeroes of random holomorphic sections of powers for positive Hermitian holomorphic line bundles from polynomial functions to entire functions. Our result can also be viewed as the analogy of Nevanlinna's First Main Theorem in the theory of the distribution of zero-sets of random entire functions.