#### Vol. 7, No. 8, 2014

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

### Junyan Zhu

Vol. 7 (2014), No. 8, 1923–1967
##### 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 $D\left(0,r\right)$.

##### Keywords
hole probability, asymptotic, SU(m+1) polynomial
##### Mathematical Subject Classification 2010
Primary: 32A60, 60D05