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Gaussian Complex Zeros on the Hole Event: The Emergence of a Forbidden Region
Author(s) -
Ghosh Subhroshekhar,
Nishry Alon
Publication year - 2019
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21800
Subject(s) - mathematics , measure (data warehouse) , gaussian , zero (linguistics) , boundary (topology) , complex plane , function (biology) , sequence (biology) , plane (geometry) , mathematical analysis , radius , combinatorics , pure mathematics , geometry , quantum mechanics , physics , linguistics , philosophy , genetics , computer security , database , evolutionary biology , computer science , biology
Abstract Consider the Gaussian entire functionF ℂ ( z ) = ∑ k = 0 ∞ξ kz kk !, z ∈ ℂ , where {ξ k } is a sequence of independent standard complex Gaussians. This random Taylor series is distinguished by the invariance of its zero set with respect to the isometries of the plane ℂ. It has been of considerable interest to study the statistical properties of the zero set, particularly in comparison to other planar point processes. We show that the law of the zero set, conditioned on the function F ℂ having no zeros in a disk of radius r and normalized appropriately, converges to an explicit limiting Radon measure on ℂ as r → ∞. A remarkable feature of this limiting measure is the existence of a large “forbidden region” between a singular part supported on the boundary of the (scaled) hole and the equilibrium measure far from the hole. In particular, this answers a question posed by Nazarov and Sodin, and is in stark contrast to the corresponding result of Jancovici, Lebowitz, and Manificat in the random matrix setting: there is no such forbidden region for the Ginibre ensemble. © 2018 Wiley Periodicals, Inc.