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IMA New Directions Short Course

Advances in Random Matrix Theory

Advances in Random Matrix Theory

June 18-29, 2012

University of Minnesota, Twin Cities | |

University of Minnesota, Twin Cities |

Random matrix theory (RMT) has seen dramatic progress in the last decade. The mathematical study of random matrices was initiated in the late 1940's, with important contributions by Wigner, and developed by mathematical physicists, most notably Dyson, Mehta, Gaudin, Pastur in the 1960's. As a mathematical discipline, progress was less intensive until the early 1990's, when Tracy and Widom made an explicit link of RMT with integrable systems. Shortly after, the theory of Riemann Hilbert problem was shown, by Deift, Zhou and co-workers, to shed light on asymptotics and in particular universality results for classes of random matrices with explicit expressions for their joint distribution of eigenvalues. Arguably the crowning achievement of these developments was the proof, by Baik, Deift and Johansson, of random matrix limits for the asymptotics of the length of the longest increasing subsequence in a random permutation. In another direction, the introduction of free probability by Voiculescu shed light on combinatorial and algebraic aspects of RMT.

In the last decade, much effort has gone into expansions of the basic theory in directions where explicit expressions and tools like orthogonal polynomials are not available. The proposed course will highlight some of these developments. The following topics will be covered, each in about 8 hours of lecture time.

- Universality in Random Matrix Theory
- Beta Ensembles
- Multi-matrix models and expansions
- Support properties for polynomials in independent matrices

Topic 1 has seen dramatic recent progress through work of Tau,Vu, Erdos, Schlein, Yau and others. Topic 2 gives an extension of classical random matrix models through tri-diagonal realizations proposed by Dumitriu and Edelman. Topic 3 deals with the combinatorial applications of random (multi)-matrix models and the identification of coefficients in ther partition functions, a topic going back to work of physicists like t'Hooft and others but now on firm mathematical ground. The last topic is influenced by work of Haagerup and collaborators, and has a strong complex-analytic flavor.

**The speakers are:**

- Greg W. Anderson (U. Minnesota) for topic #4.
- Alice Guionnet (ENS - Lyon) for topic #3.
- Balint Virag (U. Toronto) for topic #2.
- Ofer Zeitouni (U. Minnesota/Weizmann Institute) for topic #1.

In addition, shorter presentations (2-3 hours each) on material related to the main topics of the courses will be offered by guest lecturers. In order to keep abreast of current developments, the list of speakers in those presentations will be decided at a later point. We anticipate 5-6 such speakers.

The course is intended for researchers at all levels who have had prior exposure to RMT and that are interested in learning recent techniques and results in RMT.

A partial bibliography for random matrix theory is available here