# Vector equilibrium problems and their applications to random matrix models

April 2 to April 6, 2012

at the

American Institute of Mathematics, Palo Alto, California

organized by

Pavel Bleher, Alexander Its, and Arno Kuijlaars

## Original Announcement

This workshop will be devoted to the study of vector equilibrium problems and their application to the asymptotic analysis of random matrix models.

Random matrix models have numerous applications to quantum gravity, solid state physics, statistical physics, statistics, combinatorics, number theory, and mathematical biology. In recent years, impressive progress in the solution of random matrix models was based on the development of the Riemann-Hilbert approach, large deviation results, methods of integrable systems and equilibrium problems.

The analysis of interacting random matrix models, random matrix models with external source, and normal matrix models is a next important step in the development of random matrix models and their applications. These models have a clear integrable structure, but their asymptotic theory is much less developed. It is a major challenge to apply appropriate Riemann-Hilbert methods to them.

From recent advances for some of the models it became clear that the relevant Riemann-Hilbert problem is of larger size than the usual $2 \times 2$ Riemann Hilbert problem for orthogonal polynomials. To analyze the larger size Riemann-Hilbert problem the solution of a vector equilibrium problem is needed which involves a number of interacting measures, with external fields and upper constraints. One of the measures will give the limiting mean distribution of the eigenvalues in the random matrix model.

The successful analysis has been limited so far to problems with enough symmetry where one can make an a priori statement about the supports of the measures in the vector equilibrium problem. Future progress will depend on a better understanding of the role of the vector equilibrium problem in the complex plane in problems without symmetry. Then it will be an additional issue to locate the precise contours where the measures are supported.

This workshop aims at creating an environment for studying these and related problems. Researchers form the three areas of equilibrium problems from potential theory, random matrix theory, and Riemann-Hilbert problems will work together to focus on a number of specific problems. The main topics to be discussed are:

1. Equilibrium problems in the complex plane. The determination of the relevant contours in an equilibrium problem depends on identifying contours with the so-called S-property. In the extension to vector equilibrium problems one deals with several interacting measures with one or more external fields and upper constraints. We seek to develop existence and regularity results for general vector equilibrium problems in the complex plane. The combination with an upper constraint provides an additional challenge, that we wish to resolve.
2. Larger size Riemann-Hilbert problems. The vector equilibrium problems will play a role in the asymptotic analysis of larger size Riemann-Hilbert problem that have appeared recently in connection with multiple orthogonal polynomials, multi-matrix models and certain integrable systems. There are new critical phenomena in larger size Riemann-Hilbert problems that we seek to explore.
3. Random matrix models. The random matrix model with external source and the coupled matrix model lead to specific vector equilibrium problems with external fields and constraints, whose solution is related to a spectral curve. The normal matrix model has a similar structure and we intend to develop vector equilibrium problems and Riemann-Hilbert methods for this model as well. In addition, we seek a possible interpretation of the vector equilibrium problem as a rate function for a large deviation principle.

## Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.