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

Zhu, Junyan

doi:10.2140/apde.2014.7.1923

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

Junyan Zhu

Vol. 7 (2014), No. 8, 1923–1967

DOI: 10.2140/apde.2014.7.1923

Abstract

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 $r \geq 1$ , we find asymptotic formulas and the 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 $D (0, r)$ .

Keywords

hole probability, asymptotic, SU(m+1) polynomial

Mathematical Subject Classification 2010

Primary: 32A60, 60D05

Milestones

Received: 2 April 2014

Revised: 13 August 2014

Accepted: 23 September 2014

Published: 5 February 2015

Authors

Junyan Zhu
Department of Mathematics Johns Hopkins University 3400 N. Charles St. Krieger 404 Baltimore, MD 21218 United States

Vol. 7, No. 8, 2014