Distribution of the time at which N vicious walkers reach their maximal height

TL;DR

This paper derives exact distributions for the maximum height and the time it occurs for N non-intersecting Brownian motions, linking these results to stochastic growth models and directed polymers, with validation through numerical simulations.

Contribution

It provides the first exact joint distribution of the maximum and its occurrence time for vicious walkers, connecting these to growth models and polymers.

Findings

Exact joint distribution of maximum and time for vicious walkers

Connections established with growth models and directed polymers

Numerical simulations confirm analytical results

Abstract

We study the extreme statistics of N non-intersecting Brownian motions (vicious walkers) over a unit time interval in one dimension. Using path-integral techniques we compute exactly the joint distribution of the maximum M and of the time \tau_M at which this maximum is reached. We focus in particular on non-intersecting Brownian bridges ("watermelons without wall") and non-intersecting Brownian excursions ("watermelons with a wall"). We discuss in detail the relationships between such vicious walkers models in watermelons configurations and stochastic growth models in curved geometry on the one hand and the directed polymer in a disordered medium (DPRM) with one free end-point on the other hand. We also check our results using numerical simulations of Dyson's Brownian motion and confront them with numerical simulations of the Polynuclear Growth Model (PNG) and of a model of DPRM on a discrete lattice. Some of the results presented here were announced in a recent letter [J. Rambeau and G. Schehr, Europhys. Lett. 91, 60006 (2010)].