Hole probabilities of $\mathrm{SU}(m+1)$ Gaussian random polynomials

Abstract

In this paper, we study hole probabilities $P_{0,m}\left(r,N\right)$ of $SU\left(m+1\right)$ Gaussian random polynomials of degree $N$ over a polydisc ${\left(D\left(0,r\right)\right)}^m$. When $r\ge 1$, we find asymptotic formulas and the decay rate of $log{P}_{0,m}\left(r,N\right)$. In dimension one, we also consider hole probabilities over some general open sets and compute asymptotic formulas for the generalized hole probabilities $P_{k,1}\left(r,N\right)$ over a disc $D\left(0,r\right)$.