  We show that the squared maximal height of the top path among $N$
non-intersecting Brownian bridges starting and ending at the origin 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 K.
Johansson's result that the supremum of the Airy$_2$ process minus a parabola
has the Tracy-Widom GOE distribution, and as such it provides an explanation
for how this distribution arises in models belonging to the KPZ universality
class with flat initial data. The result can be recast in terms of the
probability that the top curve of the stationary Dyson Brownian motion hits an
hyperbolic cosine barrier.
