Riemann-Hilbert Two Cut Picture

Random Matrices, Random Processes
and Integrable Systems

A Short Program of the Centre de recherches mathématiques
on the campus of the Université de Montréal.

20 June - 8 July 2005



John Harnad (CRM & Concordia University)
Jacques Hurtubise (CRM & McGill University)





Purpose of the program

This program tried to emphasize the remarkable connections between two domains that a priori seem unrelated: random matrices (together with associated random processes) and integrable systems. The relations between random matrix models and the theory of classical integrable systems have long been studied. These appear mainly in the deformation theory, when parameters characterizing the measures or the domain of localization of the eigenvalues are varied. The resulting differential equations determining the partition function and correlation functions are, remarkably, of the same type as certain equations appearing in the theory of integrable systems. They may be analyzed effectively through methods based upon the Riemann-Hilbert problem of analytic function theory and by related approaches to the study of nonlinear asymptotics in the large N limit. Associated to studies of matrix models are certain stochastic processes, the "Dyson processes", and their continuum limits, which are related to the spectra of random matrix ensembles, and may also be studied by related methods.

Besides the well-known physical applications of random matrix theory, such as the Wigner-Dyson statistical approach to the distribution of high lying resonances of large nuclei, and the more recent applications to string theory and two dimensional quantum gravity, there exist further new applications under current study, such as the computation of correlation functions in supersymmetric Yang-Mills theory, and the regularization of the Laplacian growth problem of two dimensional fluid dynamics. Correlation functions between eigenvalues of random matrices also have close similarities to those in integrable quantum spin systems and many body models. There are further remarkable connections to a variety of probabilistic problems such as random words, tilings and partitions, as well as to the statistical distribution of zeros of L functions.

The program provided an opportunity for productive interactions, bringing together top experts and younger researchers beginning work in this area. The schedule consisted of two parts. There were eight extended lecture series on related topics, each of one week's duration, having a survey and pedagogical character, aimed primarily at younger researchers entering the field. The afternoon sessions were of "workshop" character, with one hour talks presented on current work in the field.


Main topics covered

Invited speakers

Lecture series speakers

Mark Adler Brandeis Univ.
Pavel Bleher Indiana Univ.-Purdue Univ. at Indianapolis (I.U.P.U.I.)
Bertrand Eynard C.E.A,. Saclay, S.Ph.T.
Alexander Its Indiana Univ.-Purdue Univ. at Indianapolis (I.U.P.U.I.)
Ken McLaughlin University of Arizona
Craig Tracy U.C. Davis
Pierre van Moerbeke Univ. Catholique de Louvain / Brandeis Univ.
Harold Widom U.C. Santa Cruz

Workshop speakers

Marco Bertola Concordia Univ., C.R.M.
Brian Conrey American Institute of Mathematics
Percy Deift Courant Inst., N.Y.U.
Philippe di Francesco C.E.A., Saclay, S.Ph.T.
Sam Howison Math. Institute, OCIAM, Oxford
Vladimir Kazakov E.N.S., Paris
Dmitri Korotkin Concordia Univ., C.R.M.
Arno Kuijlaars Katholieke Universiteit Leuven
Andrei Okounkov Princeton Univ.
Alexander Orlov Oceanology Inst., Moscow
Alexander Soshnikov U.C. Davis
Nina Snaith University of Bristol
Anton Zabrodin I.T.E.P., Moscow
Ofer Zeitouni Univ. Minn., Technion
Paul Zinn-Justin Orsay, Univ. Paris-sud
Jean-Bernard Zuber C.E.A., Saclay, S.Ph.T.


Besides the invited speakers listed above and young researchers attending the lecture series sequences, participation was encouraged by all researchers interested in this field. The schedule and facilities were organized to accommodate a total of approximately 75 participants, coming from eighteen countries around the world, over the three week period of the program, although the steady-state number in any given week was closer to 50. The workshop part of the program also included a number of contributed talks on topics relating closely to the theme of the program. Roughly half the participants were either young researchers, postdoctoral fellows or advanced graduate students, and most of these received partial financing to help cover their travel and/ or accomodation expenses.


The Lecture Series part of this program will published in the Springer CRM Series in Mathematical Physics.
A refereed Special Issue of Journal of Physics A: Mathematical and General was published in July, 2006 based on the topics covered in the workshop part program



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