Random matrices and integrable systems

March 4 - 9, 2012

Les Houches

General

Announcement

Poster

Participants

Practical details

Schedule

SECOND ANNOUNCEMENT

The school ``Random matrices and integrable systems'', in the framework of the ANR project Grandes Matrices Aleatoires will be held in the Ecole de physique des Houches from Sunday, March 4, to Friday, March 9, 2012. The conference's aim is to bring together researchers working in all fields of mathematics and theoretical physics related to random matrices.In the past few years, random matrices have seen many new developments, and became related to many other areas of mathematics and physics, in particular to growth models or to enumerative geometry. Courses will mainly focus on the relation of random matrices with integrable systems but talks of the participants will cover a much broader list of topics, and in particular those developped by the ANR program.

ORGANIZING COMMITTEE:

INVITED SPEAKERS FOR COURSES:

• M. Bertola (Concordia University)
• V. Kazakov (ENS Paris)
• A. Kuijlaars (Katholieke Universiteit Leuven)
• A. Mironov (ITEP, Moscow)

Titles/Abstracts of courses

• Marco Bertola and Arno Kuijlaars: ``Asymptotic analysis of random matrices and orthogonal polynomials''
  In the course we discuss the application of the steepest descent analysis for Riemann-Hilbert problems to problems in random matrix theory and orthogonal polynomials. The first four lectures are based on work of Deift, Kriecherbauer, McLaughlin, Venakides and Zhou from 1999. In the second part we introduce extensions of these ideas to multiple orthogonal polynomials and multi-matrix models.
• Vladimir Kazakov: ``Quantum integrability made classical''
  Integrability of a quantum system often manifests itself in the form of classical integrablility for a related, usually discrete system. Among well known examples of such relation are the fusion procedure in quantum spin chains or the discrete Y-systems for quantum sigma models in the finite volume. In all known cases, this classical integrability takes the form of an integrable discrete Hirota dynamics, with many parallels to other applications of classical integrability, such as matrix models, group characters, integrable PDE's etc. Moreover, with a certain analyticity input, the Hirota dynamics allows to efficiently study various quantum integrable systems. This point of view recently allowed to formulate an alternative approach to the solution of rational quantum spin chains and of 2D relativistic sigma-models, and culminated in the exact solution for the full spectrum of anomalous dimensions of the four-dimensional N=4 syper-Yang-Mills theory at any 'tHooft coupling, thus giving the most convincing demonstration of the AdS/CFT correspondence. I will try to explain the basics of this approach on the example of the Heisenberg spin chain, employing a rather elementary language of GL(N) characters, matrices and matrix co-derivatives. I will construct in this way Baxter's Q-operators and derive the Bethe ansatz equations. If the time permits, I will sketch out the generalisation of the formalism to the quantum 2D sigma-models.
• Andrei Mironov: ``AGT and Matrix Models''
  A review of matrix models and $\beta$-ensembles emerging when dealing with the AGT conjecture will be presented. Various properties and generalizations of these models will be discussed in detail, putting emphasis on Seiberg-Witten structures. Integrable systems behind these systems will be also considered.