Dynamical Correlations for Vicious Random Walk with a Wall

TL;DR

This paper studies a system of nonintersecting Brownian particles with a wall, deriving exact dynamical correlations, analyzing large-system asymptotics, and connecting the model to random matrix theory universality classes.

Contribution

It provides an exact evaluation of dynamical correlations for vicious Brownian walkers with a wall and links the model to parametric random matrix ensembles.

Findings

Discontinuous transitions in dynamical correlations as N approaches infinity

Exact formulas for correlations in the presence of a wall

Equivalence to a parametric random matrix model

Abstract

A one-dimensional system of nonintersecting Brownian particles is constructed as the diffusion scaling limit of Fisher's vicious random walk model. $N$ Brownian particles start from the origin at time $t=0$ and undergo mutually avoiding motion until a finite time $t=T$. Dynamical correlation functions among the walkers are exactly evaluated in the case with a wall at the origin. Taking an asymptotic limit $N \to \infty$, we observe discontinuous transitions in the dynamical correlations. It is further shown that the vicious walk model with a wall is equivalent to a parametric random matrix model describing the crossover between the Bogoliubov-deGennes universality classes.