"Maximal height of N non-intersecting Brownian excursions: from Yang-Mills theory to interfaces in disordered media"
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Non-intersecting random walkers (or ''vicious walkers'') have been studied in
various physical situations, ranging from polymer physics to wetting and melting
transitions and more recently in connection with random matrix theory or
stochastic growth processes in the Kardar-Parisi-Zhang (KPZ) universality class.
In this talk, I will present a method based on path integrals associated to free
Fermions models to study such statistical systems. I will use this method to
calculate exactly the cumulative distribution function (CDF) of the maximal
height of N non-intersecting Brownian excursions. I will show that this CDF is
identical to the partition function of 2d Yang Mills (YM) theory on a sphere
with the gauge group Sp(2N). I will show that, in the large N limit, the CDF
exhibits a third order phase transition, akin to the Douglas-Kazakov transition
found in 2d YM. I will also show that the critical behavior, close to the
transition point, is described by the Tracy-Widom distribution for $\beta = 1$,
which describes the fluctuations of the largest eigenvalue of Random Matrices
belonging to the Gaussian Orthogonal Ensemble.