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

2008/07/03 by Scott Zrebiec, Zrebiec, Scott
Mathematics · #30B20 #30C15 #60G60 #Complex Variables (math.CV) #FOS: Mathematics #Geometry and complex manifolds #Random Matrices and Applications #Stochastic processes and statistical mechanics #math.CV #msc:30B20 #msc:30C15 #msc:60G60

paper · pdf · doi:10.48550/arxiv.0807.0604

15 pages

arxiv created 2008/07/03 · openalex publication_date 2008/07/03 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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⊂\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.

Citations

• Number variance of random zeros
• The order of the decay of the hole probability for Gaussian random SU(m+1) polynomials

Related

• Topics in the Value Distribution of Random Analytic Functions
• Continuity of the asymptotics of expected zeros of fewnomials
• On the topology of Gaussian random zero sets
• The zeros of Gaussian random holomorphic functions on \Cⁿ, and hole probability
• The order of the decay of the hole probability for Gaussian random SU(m+1) polynomials