Consider N non-intersecting Brownian bridges, all starting from 0 at time t = 0 and returning to 0 at time t = 1. TheAiry$_2$ process is defined as the motion of the top path (suitably rescaled) in the large N limit. K. Johansson proved the remarkable fact that the supremum of the Airy_2 process minus a parabola has the Tracy-Widom GOE distribution from random matrix theory. In this talk, I will present a result which shows that the squared maximal height of the top path in the case of finite N is distributed as the top eigenvalue of a random matrix drawn from the Laguerre Orthogonal Ensemble. This result can be thought of as a discrete version of Johansson’s result, and provides an explanation of how the Tracy-Widom GOE distribution arises in the KPZ universality class. The result 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 Daniel Remenik.

Pontificia Universidad Católica de Chile (PUC-Chile)

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Facultad de Ciencias Físicas y Matemáticas (FCFM)

Universidad de Chile

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Santiago – Chile