First-Passage Exponents of Multiple Random Walks

TL;DR

This paper studies the decay of probabilities that the nth rightmost of N noninteracting random walks remains positive over time, revealing a family of exponents with scaling behavior and exponential decay in the large N limit.

Contribution

It introduces a detailed analysis of first-passage exponents for multiple random walks, including their scaling form and exponential decay, using cone geometry and simulations.

Findings

First-passage probabilities decay algebraically with exponents beta_n.

Exponents form a scaling function in the large N limit.

Smallest exponent decays exponentially with N.

Abstract

We investigate first-passage statistics of an ensemble of N noninteracting random walks on a line. Starting from a configuration in which all particles are located in the positive half-line, we study S_n(t), the probability that the nth rightmost particle remains in the positive half-line up to time t. This quantity decays algebraically, S_n (t) ~ t^{-beta_n}, in the long-time limit. Interestingly, there is a family of nontrivial first-passage exponents, beta_1<beta_2<...<beta_{N-1}; the only exception is the two-particle case where beta_1=1/3. In the N-->infinity limit, however, the exponents attain a scaling form, beta_n(N)--> beta(z) with z=(n-N/2)/sqrt{N}. We also demonstrate that the smallest exponent decays exponentially with N. We deduce these results from first-passage kinetics of a random walk in an N-dimensional cone and confirm them using numerical simulations. Additionally, we investigate the family of exponents that characterizes leadership statistics of multiple random walks and find that in this case, the cone provides an excellent approximation.