Maximum distributions of bridges of noncolliding Brownian paths

• Published in: arXiv.org
• Main Authors: Kobayashi, Naoki, Minami Izumi, Katori, Makoto
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 03.10.2008
• Subjects: Brownian motion; Computer simulation; Conditioning; Dependence; Distribution functions; Eigenvalues; Extreme values; Mathematical analysis; Mathematical models; Mathematics - Mathematical Physics; Matrix methods; Matrix theory; Number theory; Physics - Mathematical Physics; Physics - Soft Condensed Matter; Physics - Statistical Mechanics; Stochastic processes; Symmetry
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.0808.3635

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.