Random Matrices and Extreme Value Statistics

Monday, 22 May 2017 to Tuesday, 23 May 2017
11:00 - 15:00
Room: 401 Moscow center for continuous mathematical education

Tentative schedule:

May 22 (Monday)

11-00 - 12-30: Satya Majumdar, Top eigenvalue of a random matrix: Tracy-Widom distribution and third order phase transition  - lecture 1

14-00 - 15-30: Gregory Schehr, Non-Intersecting Brownian motions: from Random matrices to Yang-Mills theory - lecture 1

 

May 23 (Tuesday)

11-00 - 12-30: Satya Majumdar, Top eigenvalue of a random matrix: Tracy-Widom distribution and third order phase transition  - lecture 2

14-00 - 15-30: Gregory Schehr, Non-Intersecting Brownian motions: from Random matrices to Yang-Mills theory - lecture 2

 

Abstracts:

Satya Majumdar, Top eigenvalue of a random matrix: Tracy-Widom distribution and third order phase transition (two lectures)

Tracy-Widom distribution describes the probability distribution of the typical fluctuations of the top eigenvalue of a Gaussian (NxN) random matrix. Over the last decade, the same distribution has surfaced in a wide variety of problems from Kardar-Parisi-Zhang (KPZ) surface growth, directed polymer, random permutations, all the way to large N-gauge theory and wireless communications, with some of these problems having no apriori connection to random matrices. Why is the Tracy-Widom distribution so ubiquitous? In statistical physics, universality is usually accompanied by a phase transition--near a critical point often the details become completely irrelevant. So, is there an underlying phase transition associated with the Tracy-Widom distribution? In this talk, I will demonstrate that for large but finite N, indeed there is an underlying third order phase transition from a `strong' coupling to a `weak' coupling phase--the Tracy-Widom distribution turns out to be the universal crossover function between these two phases for finite but large N. Several examples of this third order phase transition will be discussed.

 

Gregory Schehr, Non-Intersecting Brownian motions: from Random matrices to Yang-Mills theory (two lectures)

Non-intersecting Brownian motions, sometimes called vicious walkers, have been widely studied in physics (e.g., polymer physics, wetting and melting transition,...) and in mathematics (e.g., combinatorics, representation theory,…). I will first explain how such instances of constrained vicious walkers (e.g. bridges or excursions) are connected to various models of random matrix theory. I will then discuss extreme value questions related to such models and show that the cumulative distribution of the global maximum of N non-intersecting Brownian excursions is related to the partition function of two-dimensional Yang-Mills theory on the sphere. In particular, it displays, in the large N limit, a third order phase transition, the so-called Douglas-Kazavov transition, akin to the third order phase transition found in other random matrix models.