Ordering of Random Walks: The Leader and the Laggard

TL;DR

This paper studies the probability that the order of N random walks on a line remains unchanged over time, using electrostatic mapping to estimate decay exponents for the leader and laggard probabilities.

Contribution

It introduces a novel electrostatic approach to analyze ordering probabilities of random walks and provides new estimates for decay exponents, especially for N=4.

Findings

For N=4, the leader probability decays as t^{-0.91342}

The laggard probability decays as t^{- ext{gamma}_N} with specific values for N=2,3

As N approaches infinity, the laggard decay exponent approaches ln(N)/N

Abstract

We investigate two complementary problems related to maintaining the relative positions of N random walks on the line: (i) the leader problem, that is, the probability {\cal L}_N(t) that the leftmost particle remains the leftmost as a function of time and (ii) the laggard problem, the probability {\cal R}_N(t) that the rightmost particle never becomes the leftmost. We map these ordering problems onto an equivalent (N-1)-dimensional electrostatic problem. From this construction we obtain a very accurate estimate for {\cal L}_N(t) for N=4, the first case that is not exactly soluble: {\cal L}_4(t) ~ t^{-\beta_4}, with \beta_4=0.91342(8). The probability of being the laggard also decays algebraically, {\cal R}_N(t) ~ t^{-\gamma_N}; we derive \gamma_2=1/2, \gamma_3=3/8, and argue that \gamma_N--> ln N/N$ as N-->oo.