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 is intended 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 is meant to provide an opportunity for productive interactions, bringing together top experts and younger researchers beginning work in this area. The schedule will consist of two parts. There will be 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 will mainly be of "workshop" character, with one hour talks presented on current work the field. The schedule will be relaxed, with no more than five talks per day, in order to permit maximum time for interactions between the participants.

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.
Bertrand Eynard C.E.A,. Saclay, S.Ph.T.
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
Gordon Semenoff University of British Columbia
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.
(* = to be confirmed)


Besides the invited speakers listed above and young researchers attending the lecture series sequences, participation is encouraged by all researchers interested in this field. The schedule and facilities will be organized so as to accommodate a total of approximately 75 participants over the three week period of the program, although it is expected that the steady-state number of participants in any given week will be closer to 50. The workshop part of the program will also include a number of contributed talks on topics relating closely to the theme of the program.

Registration fees and financial support

Financial support to help defray participation expenses has been offered to a limited number of young researchers and advanced graduate students attending the extended lecture series sessions. A registration fee of $150 (Can) will be charged to all other participants to help defray minor social and organizational costs.


The Lecture Series part of this workshop will published in the CRM Series in Mathematical Physics.
The workshop proceedings will be published, if a sufficient number of speakers choose to contribute, either in the CRM-AMS proceedings and lecture notes series or as a refereed supplementary volume of a suitable journal.



Links to further information

STM maps and information

Suggested accommodations

All participants are asked to make their own reservations, unless there is some special reason for their being unable to do so. In the latter case, please contact the CRM activities coordinator at the e-mail link provided below to ask for assistance. To be sure that accommodations are available at your preferred location, it is recommended that accommodations be booked well ahead of time.

Further information on access to the CRM, accommodations, maps

CRM Mathematical Physics Laboratory

CRM home site

Send e-mail to CRM scientific activities coordinator

Send e-mail to the organizers