Year: 2025
Type: article
Abstract: We consider the family { f L } L >0 of Gaussian analytic functions in the unit disk, distinguished by the invariance of their zero set with respect to hyperbolic isometries. Let n L ( r ) be the number of zeros of f L in a disk of radius r . We study the asymptotic probability of the rare event where there is an overcrowding of the zeros as r ↑ 1, i.e., for every L > 0, we are looking for the asymptotics of the probability ℙ[ n L ( r ) ≥ V ( r )] with V ( r ) large compared to the $${\mathbb E}[n_{L}(r)]$$ E [ n L ( r ) ] . Peres and Virág showed that for L = 1 (and only then) the zero set forms a determinantal point process, making many explicit computations possible. Curiously, contrary to the much better understood planar model, it appears that for L < 1 the exponential order of decay of the probability of overcrowding when V is close to $${\mathbb E}[n_{L}(r)]$$ E [ n L ( r ) ] is much less than the probability of a deficit of zeros.
Source: Israel Journal of Mathematics
Author: Keren Mor Waknin
Institution: Tel Aviv University
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,290
OpenAlex ID: W4410312133