American Institute of Mathematics, Palo Alto, California
Pavel Bleher, Alexander Its, and Arno Kuijlaars
This workshop, sponsored by AIM and the NSF, 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:
The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.
The deadline to apply for support to participate in this workshop has passed.
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