Department of Mathematics,
University of California San Diego
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Math 288 - Probability & Statistics
Prof. Elliot Paquette
McGill University
Kahane’s coverage question and the image of Gaussian analytic function
Abstract:
We consider the range of Gaussian analytic functions (GAF) with finite radius of convergence. We show that any unbounded GAF has dense image in the plane. We moreover show that if in addition the coefficients have sufficiently regular variances, then the image is the whole complex plane. We do this by exploiting an approximate connection between the coverage problem and spatial branching processes, analogous to the branching structure that appears in the log-correlated GAF and circular beta ensembles. This answers a long-standing open question of J.-P. Kahane, with sufficient regularity.
Joint work with Alon Nishry.
Host: Ioana Dumitriu
December 5, 2024
11:00 AM
APM 6402
Research Areas
Probability Theory****************************
