Ordering statistics of 4 random walkers on the line

TL;DR

This paper provides highly precise estimates of the decay exponents for ordering statistics of four random walkers on a line, revealing subtle correlations and improving understanding of their long-time behavior.

Contribution

The study offers the most accurate exponents for ordering probabilities of four walkers and analyzes correlations between walkers that lag jointly, extending previous knowledge.

Findings

Decay exponent for leading walker: ~0.91287850

Decay exponent for lagging walkers: ~0.30763604

Correlations between lagging walkers are subtle but significant

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 $t$; $P_{\rm lead}(t)\sim t^{-0.91287850}$, and that a particle lags to time $t$ (never assumes the lead); $P_{\rm lag}(t)\sim t^{-0.30763604}$. 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 $P_{\rm lead}(t)\sim P_{\rm lag}(t)^2$. In contrast, our results rule out the possibility that $P_{\rm lead}(t)\sim P_{\rm lag}(t)^3$ for $N=4$, though the correlations in this borderline case are tiny.