Ordering of random walks: the leader and the laggard

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 N(t) that the leftmost particle remains the leftmost as a function of time and (ii) the laggard problem, the probability 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 N(t) for N = 4, the first case that is not exactly solvable: 4(t) t??4, with ?4 = 0.91342(8). The probability of being the laggard also decays algebraically, N(t) t??N; we derive ?2 = 1/2, ?3 = 3/8, and argue that ?N ? N?1 ln N as N ? ?.

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