### Abstract

Introducing the time in random matrix ensembles, Dyson has shown that its spectrum evolves according to non-intersecting Brownian motions held together by a drift term. For large size random matrices, the universal edge, gap and bulk scalings applied to such diffusions lead to the Airy, Pearcey and Sine processes. The transition probabilities for these infinite-dimensional random processes are governed by non-linear equations, which I plan on describing.