# Random Matrices, Inverse Spectral Methods and Asymptotics (08w5017)

Arriving in Banff, Alberta Sunday, October 5 and departing Friday October 10, 2008

## Organizers

Estelle Basor (American Institute of Mathematics)

(Concordia University)

(IPhT CEA Saclay)

(Centre de recherches mathematiques, Universite de Montreal, and Concordia University)

Alexander Its (Indiana University - Purdue University Indianapolis)

(University of Arizona)

## Objectives

The objectives of this workshop will be to:
1) give an opportunity to communicate the latest developments in the directions relating most closely to inverse spectral methods and large N asymptotics, and
2) bring together the top experts in the field, as well as promising younger researchers to help stimulate new ideas and work.
In particular, the following topics are expected to figure prominently:
- Extensions of asymptotic and inverse spectral methods to the case of biorthogonal and multi-orthogonal polynomials, with applications to multi-matrix models
- New results on universality properties within non-standard classes of matrix ensembles, and in multi-matrix systems
- New results relating to random processes, including processes involving the eigenvalues of random matrices (Dyson processes) on the real line or circle, as well as deterministic growth processes involving the support domains of complex eigenvalues of Normal Matrices
- Other relations between random matrix techniques and random processes such as asymmetric exclusion processes, polynuclear growth, and asymptotics of random partitions
- New results relating the partition functions, gap probabilities and correlators to the tau function of integrable systems theory, and to the theory of isomonodromic deformations in the resonant and irregular singularities cases.

Amongst the main achievements of the past decade were:
1) The development and application of the nonlinear steepest descent method and Riemann-Hilbert method for the study of asymptotic properties in RMT and orthogonal polynomials (Deift, Fokas, Its, Kitaev, McLaughlin, Zhou)
2) The discovery that the Tracy-Widom probability distributions arising in random matrix theory are also present in the asymptotic statistical behavior of longest increasing subsequence problems in combinatorics, as well as in the limiting statistics of random tiling problems, random growth processes, interacting particle systems and queueing theory (Baik, Deift, Johansson, Tracy, Widom)
3) The discovery and proof of the universality classes of the sine, Airy and Bessel kernels (Bleher, Its, Deift, Basor, Pastur, Shcherbina)
4) The connection between matrix integrals and the enumeration of graphs on surfaces (t'Hooft, Brezin, Itzykzon, Parisi, Zuber, Eynard, Chekhov);
5) The relation of partition functions and spacing distributions to tau functions in the theory of integrable systems and isomonodromic deformations (Tracy, Widom, Its, Harnad, Bertola, Eynard)
6) The surprising coincidence of the distributions in the Gaussian Unitary Ensemble and the nontrivial zeroes of the Riemann zeta function, as well as between value distributions of $L$-functions and those of characteristic polynomials in GUE (Montgomery, Dyson, Odlyzko, Keating, Snaith, Conrey)
7) Spectral duality and asymptotics in multi-matrix models (Migdal, Matytsin, Guionnet, Zeitouni, Bertola, Eynard, Harnad)
8) The connection between large N limits, dispersionless hierarchies, critical limits and minimal CFT (Kazakov, Kostov, Dijkgraaf, Verlinde, Wiegman, Zabrodin)