Abstract
We compute the exact decay rate of the hole probabilities for $\beta $-ensembles and determinantal point processes associated with the Mittag-Leffler kernels in the complex plane. We show that the precise decay rate of the hole probabilities is determined by a solution to a variational problem from potential theory for both processes.
Highlights
Introduction and main resultsLet U be an open subset of the complex plane
The probability that U contains no points of X, a point process in the complex plane, is called hole/gap probability for U
We calculate the asymptotics of the hole probabilities for finite β-ensembles and determinantal point processes associated with the Mittag-Leffler kernels in the complex plane, which we describe
Summary
Let U be an open subset of the complex plane. The probability that U contains no points of X , a point process (see [DVJ08, p. 7]) in the complex plane, is called hole/gap probability for U. The hole probability for various point processes in the complex plane has been studied extensively in the literature. We calculate the asymptotics of the hole probabilities for finite β-ensembles and determinantal point processes associated with the Mittag-Leffler kernels in the complex plane, which we describe . Let Xn(g,β) denote the finite β-ensemble with n points in the complex plane, where β > 0 and g satisfies Assumption 1.1. Dm(zk), i
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