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Gaussian complex zeroes are not always normal: limit theorems on the disc

Jeremiah Buckley and Alon Nishry

Vol. 3 (2022), No. 3, 675–706

We study the zeroes of a family of random holomorphic functions on the unit disc, distinguished by their invariance with respect to the hyperbolic geometry. Our main finding is a transition in the limiting behaviour of the number of zeroes in a large hyperbolic disc. We find a normal distribution if the covariance decays faster than a certain critical value. In contrast, in the regime of “long-range dependence” when the covariance decays slowly, the limiting distribution is skewed. For a closely related model we emphasise a link with Gaussian multiplicative chaos.

Gaussian analytic functions, stationary point processes, Wiener chaos
Mathematical Subject Classification
Primary: 30B20, 60F05, 60G15
Secondary: 60G55
Received: 16 August 2021
Revised: 6 April 2022
Accepted: 29 April 2022
Published: 12 December 2022
Jeremiah Buckley
Department of Mathematics
King’s College London
United Kingdom
Alon Nishry
School of Mathematical Sciences
Tel Aviv University
Tel Aviv