Overcrowding for zeros of Hyperbolic Gaussian analytic functions

TL;DR

This paper investigates the probability of overcrowding of zeros in hyperbolic Gaussian analytic functions within the unit disk, revealing differences in decay rates of overcrowding versus deficit events depending on the parameter L.

Contribution

It provides asymptotic estimates for overcrowding probabilities of zeros in hyperbolic GAFs, highlighting distinct behaviors for different parameter regimes and extending understanding beyond the planar case.

Findings

Overcrowding probability decay is slower for L<1 compared to the zero deficit probability.

For L=1, the zero set forms a determinantal point process, enabling explicit calculations.

The exponential decay rate of overcrowding probabilities varies significantly with L.

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 $n_L\left(r\right)$ 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 $r\uparrow1$, i.e. for every $L>0$, we are looking for the asymptotics of the probability $\mathbb{P}\left[n_L(r)\geq V(r)\right]$ with $V\left(r\right)$ large compared to the $\mathbb{E}\left[n_L\left(r\right)\right]$. Peres and Vir\'ag 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 $\mathbb{E}\left[n_L\left(r\right)\right]$ is much less than the probability of a deficit of zeros.