Related papers: Topics in the Value Distribution of Random Analyti…
In this note we initiate the probabilistic study of the critical points of polynomials of large degree with a given distribution of roots. Namely, let f be a polynomial of degree n whose zeros are chosen IID from a probability measure mu on…
These expository lectures focus on the distribution of zeros of the Riemann zeta function. The topics include the prime number theorem, the Riemann hypothesis, mean value theorems, and random matrix models.
In this paper, we quantify the rate of convergence between the distribution of number of zeros of random trigonometric polynomials (RTP) with i.i.d. centered random coefficients and the number of zeros of a stationary centered Gaussian…
In this work we consider the inverse problem of determining the properties of a Wigner function from the set of its zeros (the nodal set). The previous state of the art of the problem is Hudson's theorem, which shows that an empty nodal set…
The aim of this paper is to introduce a new technique for calculation of observables, in particular multiplicity distributions, in various statistical ensembles at finite volume. The method is based on Fourier analysis of the grand…
We introduce a finite version of free probability for rectangular matrices that amounts to operations on singular values of polynomials. We show that we can replicate the transforms from free probability, and that asymptotically there is…
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 purpose of this note is to study asymptotic zero distribution of multivariate random polynomials as their degrees grow. For a smooth weight function with super logarithmic growth at infinity, we consider random linear combinations of…
We study the distribution of singular values of product of random matrices pertinent to the analysis of deep neural networks. The matrices resemble the product of the sample covariance matrices, however, an important difference is that the…
In the present article, we investigate the univalence property of polyanalytic functions and $\log$-$\alpha$-analytic functions. First, by using a new idea, we prove an improved lemma and the coefficient estimates for bounded polyanalytic…
We consider the class of multiple Fourier series associated with functions in the Dirichlet space of the polydisc. We prove that every such series is summable with respect to unrestricted rectangular partial sums, everywhere except for a…
We consider very general "random integers" and (attempt to) prove that many multiplicative and additive functions of such integers have limiting distributions. These integers include, for instance, the curvatures of Apollonian circle…
We study the distribution of the zeroes of the L-functions of curves in the Artin-Schreier family. We consider the number of zeroes in short intervals and obtain partial results which agree with a random unitary matrix model.
We study the asymptotic laws for the spatial distribution and the number of connected components of zero sets of smooth Gaussian random functions of several real variables. The primary examples are various Gaussian ensembles of real-valued…
In this paper we provide a detailed study on effective versions of the celebrated Bilu's equidistribution theorem for Galois orbits of sequences of points of small height in the $N$-dimensional algebraic torus, identifying the quantitative…
By making use of the familiar Mathieu series and its generalizations, the authors derive a number of new integral representations and present a systematic study of probability density functions and probability distributions associated with…
The main result of the article reads: the distribution of a continuous starting from zero local martingale whose quadratic characteristic is almost surely absolutely continuous with respect to some non-random increasing continuous function…
We study zero distribution of random linear combinations of the form $$P_n(z)=\sum_{j=0}^n\eta_jf_j(z),$$ in any Jordan region $\Omega \subset \mathbb C$. The basis functions $f_j$ are entire functions that are real-valued on the real line,…
Function approximation is a generic process in a variety of computational problems, from data interpolation to the solution of differential equations and inverse problems. In this work, a unified approach for such techniques is…
In this paper, we discuss the value-distribution of the Riemann zeta-function. The authors give some results for the discrepancy estimate and large deviations in the limit theorem by Bohr and Jessen.