Estimates on the Probability of Outliers for Real Random Bargmann-Fock functions

TL;DR

This paper investigates the probability of rare events related to the zeros of real random Bargmann-Fock functions, providing exponential decay estimates for the likelihood of large deviations in zero distributions.

Contribution

It offers new estimates for the probabilities of large deviations in the zero set volume of real random Bargmann-Fock functions, including both upper and lower bounds.

Findings

Probability of no zeros in a large cube decays exponentially with volume

Order of decay for atypically large or small zero set volume is computed

Lower bounds on decay rates are established, though not necessarily sharp

Abstract

In this paper we consider the distribution of the zeros of a real random Bargmann-Fock function of one or more variables. For these random functions we prove estimates for two types of families of events, both of which are large deviations from the mean. First, we prove that the probability there are no zeros in $[-r,r]^m\subset\R^m$ decays at least exponentially in terms of $r^m$. For this event we also prove a lower bound on the order of decay, which we do not expect to be sharp. Secondly, we compute the order of decay for the probability of families of events where the volume of the complex zero set is either much larger or much smaller then expected.