• Corpus ID: 119149740

Universality for zeros of random analytic functions

  title={Universality for zeros of random analytic functions},
  author={Zakhar Kabluchko and Dmitry Zaporozhets},
  journal={arXiv: Probability},
Let 0; 1;::: be independent identically distributed (i.i.d.) ran- dom variables such that E log(1 +j 0j) < 1. We consider random analytic functions of the form Gn(z) = 1 X k=0 kfk;nz k ; n logf(tn);n! u(t) as n!1, where u(t) is some function, we show that the measure n converges weakly to some deterministic measure which is characterized in terms of the Legendre{Fenchel transform of u. The limiting measure is universal, that is it does not depend on the distribution of the k's. This result is… 

Figures and Tables from this paper

Roots of random functions: A framework for local universality

A robust framework is developed to solve the local distribution of roots of random functions of the form F_n(z) = sum_{i=1}^n\xi_i\phi_i( z) , and derives the first local universality result for random trigonometric polynomials with arbitrary coefficients.

Roots of random functions: A general condition for local universality

We investigate the local distribution of roots of random functions of the form $F_n(z)= \sum_{i=1}^n \xi_i \phi_i(z) $, where $\xi_i$ are independent random variables and $\phi_i (z) $ are arbitrary

Roots of random functions

A robust framework is developed to solve the local distribution of roots of random functions of the form F_n(z) by reducing, via universality theorems, the calculation of the distribution of the roots and the interaction between them to the case where $\xi_i$ are gaussian.

Zero distribution of random polynomials

We study global distribution of zeros for a wide range of ensembles of random polynomials. Two main directions are related to almost sure limits of the zero counting measures and to quantitative

No zero-crossings for random polynomials and the heat equation

Consider random polynomial ∑ni=0aixi of independent mean-zero normal coefficients ai, whose variance is a regularly varying function (in i) of order α. We derive general criteria for continuity of

Topics in the Value Distribution of Random Analytic Functions

This thesis is concerned with the behavior of random analytic functions. In particular, we are interested in the value distribution of Taylor series with independent random coefficients. We begin

Random trigonometric polynomials: Universality and non-universality of the variance for the number of real roots

In this paper, we study the number of real roots of random trigonometric polynomials with iid coefficients. When the coefficients have zero mean, unit variance and some finite high moments, we show

The circular law

The circular law theorem states that the empirical spectral distribution of a n×n random matrix with i.i.d. entries of variance 1/n tends to the uniform law on the unit disc of the complex plane as

Critical points of random polynomials with independent identically distributed roots

Let $X_1,X_2,...$ be independent identically distributed random variables with values in $\C$. Denote by $\mu$ the probability distribution of $X_1$. Consider a random polynomial

Real roots of random polynomials with coefficients of polynomial growth and non-zero means: a comparison principle and applications

This paper seeks to further explore the distribution of the real roots of random polynomials with non-centered coefficients. We focus on polynomials where the typical values of their coefficients




We consider ensembles of random polynomials of the form p(z) = P N=1 ajPj where {aj} are independent complex normal random variables and where {Pj} are the orthonormal polynomials on the boundary of

The Complex Zeros of Random Polynomials

Mark Kac gave an explicit formula for the expectation of the number, vn (a), of zeros of a random polynomial, n-I Pn(z) = E ?tj, j=O in any measurablc subset Q of the reals. Here, ... ?In-I are

A Universality Property of Gaussian Analytic Functions

We consider random analytic functions defined on the unit disk of the complex plane $f(z) = \sum_{n=0}^{\infty} a_{n} X_{n} z^{n}$, where the Xn’s are i.i.d., complex-valued random variables with

On the Distribution of the Zeros and α-Values of a Random Integral Function (I)

1.1. The present paper is the outcome of an original aim, to establish, as fully as might be, the conjecture that "almost all" integral functions of finite non-zero order p behave in essentials, and

Random matrices: Universality of ESDs and the circular law

Given an n x n complex matrix A, let mu(A)(x, y) := 1/n vertical bar{1 <= i <= n, Re lambda(i) <= x, Im lambda(i) <= y}vertical bar be the empirical spectral distribution (ESD) of its eigenvalues

Roots of random polynomials whose coefficients have logarithmic tails

It has been shown by Ibragimov and Zaporozhets [In Prokhorov and Contemporary Probability Theory (2013) Springer] that the complex roots of a random polynomial Gn(z)=∑nk=0ξkzk with i.i.d.


For a regular compact set K in Cm and a measure μ on K satisfying the Bernstein-Markov inequality, we consider the ensemble PN of polynomials of degree N , endowed with the Gaussian probability

Distribution of Zeros of Random and Quantum Chaotic Sections of Positive Line Bundles

Abstract:We study the limit distribution of zeros of certain sequences of holomorphic sections of high powers MN of a positive holomorphic Hermitian line bundle L over a compact complex manifold M.

Erratum to “How many zeros of a random polynomial are real?”

In Section 4.3 of the article “How many zeros of a random polynomial are real?” by Alan Edelman and Eric Kostlan (Bull. Amer. Math. Soc. (N.S.) 32 (1) (1995), 1–37), we meant to say that the

Zeros of Gaussian Analytic Functions and Determinantal Point Processes

The book examines in some depth two important classes of point processes, determinantal processes and 'Gaussian zeros', i.e., zeros of random analytic functions with Gaussian coefficients, which share a property of 'point-repulsion', and presents a primer on modern techniques on the interface of probability and analysis.