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Bálint Virág

Bio: Bálint Virág is an academic researcher from University of Toronto. The author has contributed to research in topics: Random walk & Random matrix. The author has an hindex of 31, co-authored 121 publications receiving 4062 citations. Previous affiliations of Bálint Virág include University of Colorado Boulder & Alfréd Rényi Institute of Mathematics.


Papers
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Journal ArticleDOI
TL;DR: In this paper, the authors give a probabilistic introduction to determinantal and per-manental point processes and establish analogous representations for permanental pro- cesses, with geometric variables replacing the Bernoulli variables.
Abstract: We give a probabilistic introduction to determinantal and per- manental 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 pro- cesses, 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.

584 citations

MonographDOI
13 Oct 2009
TL;DR: 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.
Abstract: 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. These processes share a property of 'point-repulsion', where distinct points are less likely to fall close to each other than in processes, such as the Poisson process, that arise from independent sampling. Nevertheless, the treatment in the book emphasizes the use of independence: for random power series, the independence of coefficients is key; for determinantal processes, the number of points in a domain is a sum of independent indicators, and this yields a satisfying explanation of the central limit theorem (CLT) for this point count. Another unifying theme of the book is invariance of considered point processes under natural transformation groups. The book strives for balance between general theory and concrete examples. On the one hand, it presents a primer on modern techniques on the interface of probability and analysis. On the other hand, a wealth of determinantal processes of intrinsic interest are analyzed; these arise from random spanning trees and eigenvalues of random matrices, as well as from special power series with determinantal zeros. The material in the book formed the basis of a graduate course given at the IAS-Park City Summer School in 2007; the only background knowledge assumed can be acquired in first-year graduate courses in analysis and probability.

572 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown that the largest eigenvalues of the beta ensembles of random matrix theory converge in distribution to the low-lying eigen values of the random Schrodinger operator − d2 dx2 + x+ 2 √ β bx restricted to the positive half-line.
Abstract: We prove that the largest eigenvalues of the beta ensembles of random matrix theory converge in distribution to the low-lying eigenvalues of the random Schrodinger operator − d2 dx2 + x+ 2 √ β bx restricted to the positive half-line, where b ′ x is white noise. In doing so we extend the definition of the Tracy-Widom(β) distributions to all β > 0, and also analyze their tails. Last, in a parallel development, we provide a second characterization of these laws in terms of a one-dimensional diffusion. The proofs rely on the associated tridiagonal matrix models and a universality result showing that the spectrum of such models converge to that of their continuum operator limit. In particular, we show how Tracy-Widom laws arise from a functional central limit theorem.

279 citations

Journal ArticleDOI
TL;DR: The authors showed that at any location away from the spectral edge, the eigenvalues of the Gaussian unitary ensemble and its general β siblings converge to Sine β, a translation invariant point process.
Abstract: We show that at any location away from the spectral edge, the eigenvalues of the Gaussian unitary ensemble and its general β siblings converge to Sine β, a translation invariant point process. This process has a geometric description in term of the Brownian carousel, a deterministic function of Brownian motion in the hyperbolic plane.

223 citations

Journal ArticleDOI
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


Cited by
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Proceedings ArticleDOI
22 Jan 2006
TL;DR: Some of the major results in random graphs and some of the more challenging open problems are reviewed, including those related to the WWW.
Abstract: We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

7,116 citations

Book ChapterDOI
01 Jan 1998
TL;DR: In this paper, the authors explore questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties, using diffusion processes as a model of a Markov process with continuous sample paths.
Abstract: We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator.

2,446 citations

Journal ArticleDOI
TL;DR: This tutorial article surveys some of these techniques based on stochastic geometry and the theory of random geometric graphs, discusses their application to model wireless networks, and presents some of the main results that have appeared in the literature.
Abstract: Wireless networks are fundamentally limited by the intensity of the received signals and by their interference. Since both of these quantities depend on the spatial location of the nodes, mathematical techniques have been developed in the last decade to provide communication-theoretic results accounting for the networks geometrical configuration. Often, the location of the nodes in the network can be modeled as random, following for example a Poisson point process. In this case, different techniques based on stochastic geometry and the theory of random geometric graphs -including point process theory, percolation theory, and probabilistic combinatorics-have led to results on the connectivity, the capacity, the outage probability, and other fundamental limits of wireless networks. This tutorial article surveys some of these techniques, discusses their application to model wireless networks, and presents some of the main results that have appeared in the literature. It also serves as an introduction to the field for the other papers in this special issue.

1,893 citations

Book
21 Dec 2009
TL;DR: The theory of random matrices plays an important role in many areas of pure mathematics and employs a variety of sophisticated mathematical tools (analytical, probabilistic and combinatorial) as mentioned in this paper.
Abstract: The theory of random matrices plays an important role in many areas of pure mathematics and employs a variety of sophisticated mathematical tools (analytical, probabilistic and combinatorial). This diverse array of tools, while attesting to the vitality of the field, presents several formidable obstacles to the newcomer, and even the expert probabilist. This rigorous introduction to the basic theory is sufficiently self-contained to be accessible to graduate students in mathematics or related sciences, who have mastered probability theory at the graduate level, but have not necessarily been exposed to advanced notions of functional analysis, algebra or geometry. Useful background material is collected in the appendices and exercises are also included throughout to test the reader's understanding. Enumerative techniques, stochastic analysis, large deviations, concentration inequalities, disintegration and Lie algebras all are introduced in the text, which will enable readers to approach the research literature with confidence.

1,289 citations

Journal ArticleDOI
TL;DR: In this article, the authors studied a random Groeth model in two dimensions closely related to the one-dimensional totally asymmetric exclusion process and showed that shape fluctuations, appropriately scaled, converges in distribution to the Tracy-Widom largest eigenvalue distribution for the Gaussian Unitary Ensemble.
Abstract: We study a certain random groeth model in two dimensions closely related to the one-dimensional totally asymmetric exclusion process. The results show that the shape fluctuations, appropriately scaled, converges in distribution to the Tracy-Widom largest eigenvalue distribution for the Gaussian Unitary Ensemble.

1,031 citations