Random zero sets for Fock type spaces

TL;DR

This paper investigates the conditions under which randomly distributed points form zero sets in weighted Fock spaces, identifying critical densities and probabilistic criteria for these sets to be uniqueness sets.

Contribution

It introduces probabilistic methods to analyze zero sets in Fock spaces, establishing conditions and critical densities for random zero sets to occur almost surely.

Findings

Identifies conditions on sequences for zero sets in Fock spaces.

Determines the critical density of sequences relative to weights.

Establishes almost sure properties of random zero sets.

Abstract

Given a nondecreasing sequence $\Lambda=\{\lambda_n>0\}$ such that $\displaystyle\lim_{n\to\infty} \lambda_n=\infty,$ we consider the sequence $\mathcal N_\Lambda:=\left\{\lambda_ne^{i\theta_n},n\in\,\mathbb N\right\}$, where $\theta_n$ are independent random variables uniformly distributed on $[0,2\pi].$ We discuss the conditions on the sequence $\Lambda$ under which $\mathcal N_\Lambda$ is a zero set (a uniqness set) of a given weighted Fock space almost surely. The critical density of the sequence $\Lambda$ with respect to the weight is found.