Bernoulli Congress July 27-31, Barcelona.

For presenters of the "Random Matrices and Related Processes" sessions

• Accommodation
• Travel Schedules
• Session Timings
• Abstracts

Please e-mail  Raj Rao (raj@mit.edu) with any corrections/updates

(Last Updated: Friday, June 16th, 12:30 pm EST)

Accommodation

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Travel Schedules 

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Session Timings (Wednesday, July 28th)

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Abstracts 

Alan Edelman: Advances in Random Matrix Theory

In addition to the steady mathematical advances in random matrix theory, our ability to compute distributions and apply the results continues to grow. In this talk, we will survey some important developments in this direction.

Some examples include the polynomial method, new results on the tails of condition numbers of random matrices, the ability to numerically compute the distributions of the smallest and largest eigenvalues and very importantly the ability to numerically compute zonal and Jack polynomials, and hypergeometric functions of matrix argument.

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Raj Rao: The polynomial method for random matrices

The empirical distribution function (e.d.f.) of random matrices with real eigenvalues is of particular interest to many research communities. For infinitely large random matrices, in many interesting cases, the e.d.f. can be obtained from the solution of a bivariate polynomial equation in the probability space variable z and the associated Stieltjes transform variable m(z), such that L(z,m(z))=0. The Marcenko-Pastur  theorem and the R and S transforms in free probability, for example, allow us to obtain such bivariate polynomials. We show that this bivariate polynomial is the most natural representation for random matrices in many instances. Our evidence for this hypothesis is our observation that if we perform simple transformations on the random matrix, then the transformations in L(z,m(z)) can be represented very simply. We present these transformations as a mathematical tool as well as a computational realization that simplifies and extends researchers abilities to get answers to complicated random matrix questions. 

This is joint work with Alan Edelman.

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Antonia Tulino: The Shannon transform in Random Matrix Theory

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Olivier Leveque: Random matrices and communication networks

We intend to present a new result concerning the capacity of ad-hoc wireless networks, that is, decentralized communication networks with no fixed infrastructure that helps relaying communications. It has been shown recently (Gupta-Kumar 2000)  that the capacity of such networks does not scale with the number  of users in the network. However, the assumptions made in order to establish this result are not of information-theoretic nature,  since it is assumed that interference in the network is treated as noise. Our aim is to recover the result of Gupta and Kumar without any assumptions on the way communications take place.

 The approach we take leads us to the study of eigenvalues of  large random matrices. In various situations, we are able to recover Gupta and Kumar's result, but the most general case is still open.

This is joint work with Emre Telatar and David Tse.

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Alexander Soshnikov

Session organizer

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Alexei Borodin: Continuous time Markov chains related to Plancherel measure

Consider the standard Poisson process in the first quadrant of the Euclidean plane, and for any point (u,v) of this quadrant take the Young diagram obtained by applying the Robinson-Schensted correspondence to the intersection of the Poisson point configuration with the rectangle with vertices (0,0), (u,0), (u,v), (0,v). It is known that the distribution of the random Young diagram thus obtained is the poissonized Plancherel measure with parameter uv.

We show that for (u,v) moving along any southeast-directed curve in the quadrant, these Young diagrams form a Markov chain with continuous time. We also describe these chains in terms of jump rates. Our main result is the computation of  the dynamical correlation functions of such Markov chains and their bulk and edge scaling limits.

This is a joint work with Grigori Olshanski.

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Craig Tracy: Differential Equations for Dyson Processes

We call Dyson process any process on ensembles of matrices in which the entries undergo diffusion. The original Dyson process is what we call the Hermite process. Scaling the Hermite process at the edge leads to the Airy process and in the bulk to the sine process. Similarly we define a Bessel process by scaling the Laguerre process at the hard edge.

For a given Dyson process one is interested in the induced process on the eigenvalues, $\lambda_j(\tau)$; and in particular, the process defined by the largest eigenvalue $\lambda_{max}(\tau)$. and in particular, the process defined by the largest eigenvalue $\lambda_{max}(\tau)$. It is known that the finite dimensional distributions $P(\lambda_{max}(\tau_1)\le \xi_1,\ldots,\lambda_{max}(\tau_m)\le \xi_m)$ are expressible in terms of a Fredholm determinat of an operator with $m\times m$ matrix kernel. For $m=1$ earlier results of the authors show that these Fredholm determinants can be expressed in terms of solutions to certain Painlev\'e equations.

We show that for general $m$ the finite dimensional distributions are expressible in terms of solutions to a system of total partial differential equations with the $\xi_j$ the independent variables. For $m=2$ a different set of PDEs were found for the Hermite and Airy process by Adler and van Moerbeke . 

In this lecture we give an overview of these results as well as discussing the significance for growth processes. 

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Ioana Dumitriu: MOPs -- A Maple Library for Multivariate Orthogonal Polynomials 
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Many problems in statistics, physics, and engineering make use of random matrix theory, and require computations of eigenvalue statistics (moments of the determinant, powers of the trace, extremal eigenvalue distributions) for the $\beta$-Hermite, Laguerre and Jacobi ensembles (which include the GOE, GUE, GSE, Wishart real and complex,  etc). Such computations often involve either evaluating $\beta$-dependent (and messy) integrals over subsets of $R^n$. 

Using multivariate orthogonal polynomial theory, we have written and implemented in Maple a set of codes which provide a unified (valid for all $\beta$) way of dealing with many such computations, for these three classical types of random matrix ensembles. These codes are fast (in a relative sense, as the complexity of the problem is super-polynomial) and have the advantage of working both symbolically and numerically. 

We will present the ideas behind the codes and the computations they involve, and exemplify the performance of MOPs in a few cases.

This is joint work with Gene Shuman and Alan Edelman (MIT). 

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