Maximum distributions of bridges of noncolliding Brownian paths

TL;DR

This paper analyzes the maximum distribution of noncolliding Bessel bridges, connecting the problem to random matrix theory, and provides exact results for the outermost path along with numerical insights for inner paths.

Contribution

It derives the exact distribution of the maximum value for the outermost path in a system of noncolliding Bessel bridges and links the process to the eigenvalues of matrix-valued Brownian bridges in symmetry class C.

Findings

Exact distribution function for the outermost path's maximum value.

Numerical results on maximum-value distributions for inner paths.

Connection established between noncolliding Bessel bridges and random matrix theory.

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 $t \in (0,1)$ 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 $2N \times 2N$ 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.