Ordering of random walks: the leader and the laggard

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.