Ordering statistics of four random walkers on a line.

Abstract

We study the ordering statistics of four random walkers on the line, obtaining a much improved estimate for the long-time decay exponent of the probability that a particle leads to time t, P_{lead}(t)∼t^{-0.91287850}, and that a particle lags to time t (never assumes the lead), P_{lag}(t)∼t^{-0.30763604}. Exponents of several other ordering statistics for N=4 walkers are obtained to eight-digit accuracy as well. The subtle correlations between n walkers that lag jointly, out of a field of N, are discussed: for N=3 there are no correlations and P_{lead}(t)∼P_{lag}(t)^{2}. In contrast, our results rule out the possibility that P_{lead}(t)∼P_{lag}(t)^{3} for N=4, although the correlations in this borderline case are tiny.