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Dynamics & probability: Alon Nishry (U. Michigan): Gaussian complex zeros on the hole event: the emergence of a forbidden region | Einstein Institute of Mathematics

Dynamics & probability: Alon Nishry (U. Michigan): Gaussian complex zeros on the hole event: the emergence of a forbidden region

Date: 
Tue, 03/01/201714:00-15:00
Location: 
Manchester building, Hebrew University of Jerusalem, (Room 209)
Consider the Gaussian Entire Function (GEF) whose Taylor coefficients are independent complex-valued Gaussian variables, and the variance of the k-th coefficient is 1/k!. This random Taylor series is distinguished by the invariance of its zero set with respect to the isometries of the complex plane.
I will show that the law of the zero set, conditioned on the GEF having no zeros in a disk of radius r, and properly normalized, converges to an explicit limiting Radon measure in the plane, as r goes to infinity. 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. This answers a question posed by Nazarov and Sodin, and is in stark contrast to the corresponding result known to hold in the random matrix setting, where such a gap does not appear.
The talk is based on a joint work with S. Ghosh.