Extreme value distributions of noncolliding diffusion processes

TL;DR

This paper studies the extreme value distributions of noncolliding diffusion processes, connecting them to random matrix eigenvalues, and provides explicit determinantal and pfaffian formulas involving special functions.

Contribution

It introduces four types of noncolliding diffusion processes with inhomogeneous time dependence and derives their extreme value distributions using determinantal and pfaffian representations.

Findings

Distribution functions expressed as determinants and pfaffians

Connections established between particle distributions and random matrix eigenvalues

Explicit formulas involve special functions

Abstract

Noncolliding diffusion processes reported in the present paper are $N$-particle systems of diffusion processes in one-dimension, which are conditioned so that all particles start from the origin and never collide with each other in a finite time interval $(0, T)$, $0 < T < \infty$. We consider four temporally inhomogeneous processes with duration $T$, the noncolliding Brownian bridge, the noncolliding Brownian motion, the noncolliding three-dimensional Bessel bridge, and the noncolliding Brownian meander. Their particle distributions at each time $t \in [0, T]$ are related to the eigenvalue distributions of random matrices in Gaussian ensembles and in some two-matrix models. Extreme values of paths in $[0, T]$ are studied for these noncolliding diffusion processes and determinantal and pfaffian representations are given for the distribution functions. The entries of the determinants and pfaffians are expressed using special functions.