Showing papers by "Bálint Virág published in 2005"
TL;DR: In this paper, it was shown that the zero set of the random power series f(z) = P anz n with i.i.d. complex Gaussian coefficientsan.
Abstract: Consider the zero set of the random power series f(z) = P anz n with i.i.d. complex Gaussian coefficientsan. We show that these zeros form a determinantal process: more precisely, their joint intensity can be written as a minor of the Bergman kernel. We show that the number of zeros of f in a disk of radius r about the origin has the same distribution as the sum of independent {0,1}-valued random variables Xk, where P(Xk = 1) = r 2k . Moreover, the set of absolute values of the zeros of f has the
188 citations
TL;DR: This paper showed that the Basilica group is amenable and thus answering an open question of Grigorchuk and Żuk [9] and showed that it is possible to separate the class of amenable groups from the closure of subexponentially growing groups under the operations of group extension and direct limits.
Abstract: We use random walks to show that the Basilica group is amenable and thus answering an open question of Grigorchuk and Żuk [9]. Our results separate the class of amenable groups from the closure of subexponentially growing groups under the operations of group extension and direct limits; these classes are separated even within the realm of finitely presented groups.
119 citations
TL;DR: In this article, the authors give a probabilistic introduction to determinantal and permanental point processes, and establish analogous representations for permanental processes, with geometric variables replacing the Bernoulli variables.
Abstract: We give a probabilistic introduction to determinantal and permanental point processes. Determinantal processes arise in physics (fermions, eigenvalues of random matrices) and in combinatorics (nonintersecting paths, random spanning trees). They have the striking property that the number of points in a region $D$ is a sum of independent Bernoulli random variables, with parameters which are eigenvalues of the relevant operator on $L^2(D)$. Moreover, any determinantal process can be represented as a mixture of determinantal projection processes. We give a simple explanation for these known facts, and establish analogous representations for permanental processes, with geometric variables replacing the Bernoulli variables. These representations lead to simple proofs of existence criteria and central limit theorems, and unify known results on the distribution of absolute values in certain processes with radially symmetric distributions.
11 citations