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 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 p vicious
walkers with a wall (excursions) and without a wall (bridges). In the case
of excursions, I will show that the CDF is identical to the partition
function of 2d Yang Mills theory on a sphere with the gauge group Sp(2p).
Taking advantage of a large p analysis achieved in that context, I will
show that the CDF, properly shifted and scaled, converges to the
Tracy-Widom distribution for $\beta = 1$, which describes the fluctuations
of the largest eigenvalue of Random Matrices in the Gaussian Orthogonal
Ensemble.