Ordering statistics of 4 random walkers on the line

Abstract

We study the ordering statistics of 4 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 $t$; $P_{\mathrm{lead}}(t)\sim t^{-0.91287850}$, and that a particle lags to time $t$ (never assumes the lead); $P_{\mathrm{lag}}(t)\sim t^{-0.30763604}$. Exponents of several other ordering statistics for $N=4$ walkers are obtained to 8 digits accuracy as well. The subtle correlations between $n$ walkers that lag {\em jointly}, out of a field of $N$, are discussed: For $N=3$ there are no correlations and $P_{\mathrm{lead}}(t)\sim P_{\mathrm{lag}}(t{)}^2$. In contrast, our results rule out the possibility that $P_{\mathrm{lead}}(t)\sim P_{\mathrm{lag}}(t{)}^3$ for $N=4$, though the correlations in this borderline case are tiny.