Non-Intersecting Brownian motions: from Random matrices to Yang-Mills theory – lecture 1

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.