Bao Nguyen: Extreme statistics of non-intersecting Brownian motions and LOE

Abstract: We review the system of non-intersecting Brownian motions. This model, on the one hand, shows scaling behaviors belong to the Kardar-Parisi-Zhang (KPZ) universality class while on the other hand it is intimately related to eigenvalues of Hermitian random matrices subjected to Dyson's Brownian motion. We will describe a formula which expresses the probability that the top path of non-intersecting Brownian motions lies below a given curve as Fredholm determinant of a certain "path-integral" kernel, and show some applications to the extreme statistics of the model.