A generalised Airy distribution function for the accumulated area swept by $N$ vicious Brownian paths

TL;DR

This paper derives exact distribution functions for the area swept by multiple vicious Brownian paths using a generalized Airy distribution, linking stochastic processes with Random Matrix Theory and confirming results via simulations.

Contribution

It introduces a generalized Airy distribution function for the accumulated area of N vicious Brownian paths, connecting stochastic processes with Random Matrix Theory.

Findings

Exact distribution functions derived for N vicious Brownian paths

The generalized Airy distribution involves the Vandermonde determinant of Airy roots

Monte Carlo simulations confirm the theoretical results

Abstract

In this work exact expressions for the distribution function of the accumulated area swept by reunions and meanders of $N$ vicious Brownian particles up to time $T$ are derived. The results are expressed in terms of a generalised Airy distribution function, containing the Vandermonde determinant of the Airy roots. By mapping the problem to an Random Matrix Theory ensemble we are able to perform Monte Carlo simulations finding perfect agreement with the theoretical results.