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.Comment: 7 pages, 4 figures, 2-column revtex 4 forma
