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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…
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 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…
We consider eigenvalues of a product of n non-Hermitian, independent random matrices. Each matrix in this product is of size N\times N with independent standard complex Gaussian variables. The eigenvalues of such a product form 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…
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…
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…
Near-Gaussian probability densities are common in many important physical applications. Here we develop an asymptotic expansion methodology for computing entropic functionals for such densities. The expansion proposed is a close relative of…
We compute the exact asymptotics for the cumulants of linear statistics associated with the zeros counting measure of a large class of real Gaussian processes. Precisely, we show that if the underlying covariance function is regular and…
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…
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…
Asymptotic expansions are derived for Gegenbauer (ultraspherical) polynomials for large order $n$ that are uniformly valid for unbounded complex values of the argument $z$, including the real interval $0 \leq z \leq 1$ in which the zeros in…
We consider the uniform asymptotic expansion for the Gauss hypergeometric function \[F(a+\epsilon\lambda,m;c+\lambda;x),\qquad \lambda\to+\infty\] for $x<1$ and positive integer $m$ when the parameter $\epsilon>1$ and the constants $a$ and…
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 give overcrowding estimates for the Sine_beta process, the bulk point process limit of the Gaussian beta-ensemble. We show that the probability of having at least n points in a fixed interval is given by $e^{-\frac{\beta}{2} n^2…
We consider random trigonometric polynomials of the form \[ f_n(t):=\frac{1}{\sqrt{n}} \sum_{k=1}^{n}a_k \cos(k t)+b_k \sin(k t), \] where $(a_k)_{k\geq 1}$ and $(b_k)_{k\geq 1}$ are two independent stationary Gaussian processes with the…
Let $K_n$ be the convex hull of i.i.d. random variables distributed according to the standard normal distribution on $\R^d$. We establish variance asymptotics as $n \to \infty$ for the re-scaled intrinsic volumes and $k$-face functionals of…
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…
Gaussian quadrature rules are a classical tool for the numerical approximation of integrals with smooth integrands and positive weight functions. We derive and expicitly list asymptotic expressions for the points and weights of Gaussian…
Let $f:\mathbb{R}^d \to \mathbb{R}^k$ be a smooth centered stationary Gaussian field and $\mathcal{B} \subset \mathbb{R}^d$ be a bounded Borel set. In this paper, we determine the asymptotics as $R \to \infty$ of all the cumulants of the…