Related papers: Overcrowding for zeros of Hyperbolic Gaussian anal…
We consider the point process of zeroes of certain Gaussian analytic functions and find the asymptotics for the probability that there are more than m points of the process in a fixed disk of radius r, as m-->infinity. For the Planar…
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…
The zero set of the hyperbolic Gaussian analytic function is a random point process in the unit disc whose distribution is invariant under automorphisms of the disc. We study the variance of the number of points in a disc of increasing…
The "hole probability" that the zero set of the time dependent planar Gaussian analytic function f(z,t) = sum_(n=0)^infty a_n(t) z^n/sqrt(n!), where a_n(t) are i.i.d. complex valued Ornstein-Uhlenbeck processes, does not intersect a disk of…
We study the probability that a real stationary Gaussian process has at least $\eta T$ zeros in $[0,T]$ (overcrowding), or at most this number (undercrowding). We show that if the spectral measure of the process is supported on $\pm[B,A]$,…
The dominant theme of this thesis is that random matrix valued analytic functions, generalizing both random matrices and random analytic functions, for many purposes can (and perhaps should) be effectively studied in that level of…
We investigate radial statistics of zeros of hyperbolic Gaussian Analytic Functions (GAF) of the form $\varphi (z) = \sum_{k\ge 0} c_k z^k$ given that $|\varphi (0)|^2=t$ and assuming coefficients $c_k$ to be independent standard complex…
Assuming certain conditions on the spectral measures of centered stationary Gaussian processes on $\mathbb{R}$ (or ${\mathbb{R}}^2$), we show that the probability of the event that their zero count in an interval (resp., nodal length in a…
We consider three models (elliptic, flat and hyperbolic) of Gaussian random analytic functions distinguished by invariance of their zeroes distribution. Asymptotic normality is proven for smooth functionals (linear statistics) of the set of…
Given a Gaussian analytic function $f_L$ of intesity $L$ in the unit ball of $\mathbb C^n$, $n\geq 2$, consider its (random) zero variety $Z(f_L)$. We study the variance of the $(n-1)$-dimensional volume of $Z(f_L)$ inside a…
This is the first installment in a series of papers devoted to examining certain aspects of the asymptotic value distribution and distribution of zeros manifested by members of a broad class of linear combinations of L-functions in the…
We study the hole probability of Gaussian random entire functions. More specifically, we work with the flat model (the zero set of this function has a distribution which is invariant with respect to the plane isometries). A hole is the…
We study hyperbolic Gaussian analytic functions in the unit polydisk of $\mathbb C^n$. Following the scheme previously used in the unit ball we first study the asymptotics of fluctuations of linear statistics as the directional intensities…
We study the weak asymptotic behavior of the zeros of a family of a certain class of (generalized) hypergeometric polynomials, using the associated hypergeometric differential equation, as the parameters go to infinity. We describe the…
We study the zeros of random power series with stationary complex Gaussian coefficients, whose spectral measure is absolutely continuous. We analyze the precise asymptotic behavior of the radial density of zeros near the boundary of the…
We study a family of random Taylor series $$F(z) = \sum_{n\ge 0} \zeta_n a_n z^n$$ with radius of convergence almost surely $1$ and independent identically distributed complex Gaussian coefficients $(\zeta_n)$; these Taylor series are…
We study a new type of extremal problem in complex analysis, referred to as "geometric zero packing", which is the hyperbolic analogue of a problem considered by Abrikosov in the 1950s concerning Bose-Einstein condensates. We relate the…
Let $f:\mathbb{R} \to \mathbb{R}$ be a stationary centered Gaussian process. For any $R>0$, let $\nu_R$ denote the counting measure of $\{x \in \mathbb{R} \mid f(Rx)=0\}$. In this paper, we study the large $R$ asymptotic distribution of…
A classical fact of the theory of almost periodic functions is the existence of their asymptotic distributions. In probabilistic terms, this means that if $f$ is a Besicovitch almost periodic function and $V$ is a random variable uniformly…
We study sampling properties of the zero set of the Gaussian entire function on Fock spaces. Firstly, we relax Seip and Wallst\'en's density and separation conditions for sampling sets on Fock spaces to obtain weighted inequalities for sets…