Hole probabilities of SU(m+1) Gaussian random polynomials

In this paper, we study hole probabilities $P_{0,m}(r,N)$ of SU(m+1) Gaussian random polynomials of degree $N$ over a polydisc $(D(0,r))^m$. When $r\geq1$, we find asymptotic formulas and decay rate of $\log{P_{0,m}(r,N)}$. In dimension one, we also consider hole probabilities over some general open sets and compute asymptotic formulas for the generalized hole probabilities $P_{k,1}(r,N)$ over a disc $D(0,r)$.