Maximum distributions of bridges of noncolliding Brownian paths

Abstract

The 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 the 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 ∈ (0, 1) are studied to characterize the statistics of random patterns of the repulsive paths on the spatio-temporal 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 2N ×2N 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 with the random matrix theory, representation theory of symmetry, and the number theory.

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Ornstein–Uhlenbeck diffusion of hermitian and non-hermitian matrices—unexpected links

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