Maximum distributions of bridges of noncolliding Brownian paths

One-dimensional Brownian motion starting from the origin at time t=0, conditioned to return to the origin at time t=1 and to stay positive during time interval 0 < t < 1, is called the Bessel bridge with duration 1. We consider an N-particle system of such Bessel bridges conditioned never to collide with each other in 0 < t < 1, which is the continuum limit of the vicious walk model in watermelon configuration with a wall. Distributions of maximum values of paths attained in the time interval t is an element of(0,1) are studied to characterize the statistics of random patterns of the repulsive paths on the spatiotemporal plane. For the outermost path, the distribution function of maximum value is exactly determined for general N. We show that the present N-path system of noncolliding Bessel bridges is realized as the positive-eigenvalue process of the 2Nx2N matrix-valued Brownian bridge in the symmetry class C. Using this fact, computer simulations are performed and numerical results on the N dependence of the maximum-value distributions of the inner paths are reported. The present work demonstrates that the extreme-value problems of noncolliding paths are related to random matrix theory, the representation theory of symmetry, and number theory.