One of the main features of models in the Kardar-Parisi-Zhang (KPZ) universality class is the fact that, for some special classes of initial data, their asymptotic fluctuations are described by objects coming from random matrix theory (RMT). This connection is well understood in the case of curved initial data, which lead to Tracy-Widom GUE asymptotics.

In the case of flat initial data, on the other hand, the appearance of Tracy-Widom GOE asymptotics is far from being understood. In this talk I will provide an explanation for how the connection arises, by focusing on a system of N non-intersecting Brownian bridges, one of the simplest models in the KPZ class.

I will present a result which shows that the squared maximal height of the top path in this system is distributed as the top eigenvalue of a (finite) random matrix drawn from the Laguerre Orthogonal Ensemble, which in turn is known to converge (under the right scaling) to the Tracy-Widom GOE distribution.

This result can be thought of as a discrete version of K. Johansson’s result that the supremum of the Airy2 process minus a parabola has the Tracy-Widom GOE distribution, and can also be recast in terms of the probability that the top curve of the stationary Dyson Brownian motion hits an hyperbolic cosine barrier. This is joint work with Gia Bao Nguyen.