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

TL;DR

This paper investigates the decay rate of the hole probability for Gaussian random SU(m+1) polynomials, establishing exponential bounds in terms of the polynomial degree and deriving general volume deviation estimates for zero sets.

Contribution

It provides precise exponential bounds on the hole probability for high-degree Gaussian SU(m+1) polynomials and introduces general estimates for zero set volume deviations.

Findings

Hole probability decays exponentially with degree N^{m+1}

Established upper and lower bounds for hole probability

Derived general results on zero set volume deviations

Abstract

We show that for Gaussian random SU(m+1) polynomials of a large degree N the probability that there are no zeros in the disk of radius r is less than $e^{-c_{1,r} N^{m+1}}$, and is also greater than $e^{-c_{2,r} N^{m+1}}$. Enroute to this result, we also derive a more general result: probability estimates for the event where the volume of the zero set of a random polynomial of high degree deviates significantly from its mean.