Maximum Distance Between the Leader and the Laggard for Three Brownian Walkers

TL;DR

This paper analyzes the maximum distance between a leader and laggards among three Brownian walkers, deriving the probability distribution and its asymptotic behavior for arbitrary diffusion constants.

Contribution

It provides an exact calculation of the distribution of maximum distance and its amplitude for three Brownian walkers with arbitrary diffusion rates.

Findings

Derived the probability distribution P(m|y_2,y_3) for maximum distance.

Established the asymptotic decay form with exponent δ depending on diffusion constants.

Calculated the amplitude A(y_2,y_3) exactly.

Abstract

We consider three independent Brownian walkers moving on a line. The process terminates when the left-most walker (the `Leader') meets either of the other two walkers. For arbitrary values of the diffusion constants D_1 (the Leader), D_2 and D_3 of the three walkers, we compute the probability distribution P(m|y_2,y_3) of the maximum distance m between the Leader and the current right-most particle (the `Laggard') during the process, where y_2 and y_3 are the initial distances between the leader and the other two walkers. The result has, for large m, the form P(m|y_2,y_3) \sim A(y_2,y_3) m^{-\delta}, where \delta = (2\pi-\theta)/(\pi-\theta) and \theta = cos^{-1}(D_1/\sqrt{(D_1+D_2)(D_1+D_3)}. The amplitude A(y_2,y_3) is also determined exactly.