Non-intersecting Brownian bridges and the Laguerre Orthogonal Ensemble

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 pre-asymptotic version of K. Johansson's result (Comm. Math. Phys. 242 (2003) 277-329) 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. Our proof is based on a formula, derived in (Ann. Inst. Henri Poincare B, Calc. Probab. Stat. 51 (2015) 28-58), for the probability that Dyson Brownian motion stays below a curve on a finite interval, which is given in terms of the Fredholm determinant of a certain "path-integral" kernel.