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 L-N(t) that the leftmost particle remains the leftmost as a function of time and (ii) the laggard problem, the probability 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 L-N(t) for N = 4, the first case that is not exactly solvable: L-4(t) alpha t(-beta4), with beta(4) = 0.913 42(8). The probability of being the laggard also decays algebraically, R-N(t) alpha t(-gammaN); we derive gamma(2) = 1/2, gamma(3) = 3/8, and argue that gamma(N) --> N-1 ln N as N --> infinity.