Moscow, May 22-23, 2017

Organizers:

- Skoltech,
- Interdisciplinary Scientific Center Poncelet

**May 22 -23, 2017 **

11:00 – 15:30

Room: 401, Independent University of Moscow (Bolshoy Vlasyevskiy Pereulok 11),

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