Dyson non-intersecting Brownian motions and Airy processes

Abstract:

Consider n non-intersecting Brownian particles on the real line, leaving from and forced to the origin. In this talk, I discuss the statistical fluctuations of the largest particles, when the number n of particles tend to infinity and after some space-time rescaling. It will be shown how the presence of a finite number of outliers will influence these statistical fluctuations and how it will lead to phase transitions; outliers are Brownian particles which eventually depart from a point and end up at point, different from the origin.