Vicious Walkers in a Potential

TL;DR

This paper derives exact asymptotic survival probabilities for N vicious walkers in one-dimensional attractive potentials, including square-well and harmonic potentials, using backward Fokker-Planck equations and mappings to known problems.

Contribution

It provides a unified method to compute survival probabilities for vicious walkers in various potentials, extending previous results and introducing new cases with reflecting boundaries.

Findings

Exact asymptotic forms for survival probability Q(x,t) in attractive potentials.

Mapping of vicious walkers in zero potential to harmonic potential results.

New result for semi-infinite line with reflecting boundary: Q(x,t)  t^{-N(N-1)/2}.

Abstract

We consider N vicious walkers moving in one dimension in a one-body potential v(x). Using the backward Fokker-Planck equation we derive exact results for the asymptotic form of the survival probability Q(x,t) of vicious walkers initially located at (x_1,...,x_N) = x, when v(x) is an arbitrary attractive potential. Explicit results are given for a square-well potential with absorbing or reflecting boundary conditions at the walls, and for a harmonic potential with an absorbing or reflecting boundary at the origin and the walkers starting on the positive half line. By mapping the problem of N vicious walkers in zero potential onto the harmonic potential problem, we rederive the results of Fisher [J. Stat. Phys. 34, 667 (1984)] and Krattenthaler et al. [J. Phys. A 33}, 8835 (2000)] respectively for vicious walkers on an infinite line and on a semi-infinite line with an absorbing wall at the origin. This mapping also gives a new result for vicious walkers on a semi-infinite line with a reflecting boundary at the origin: Q(x,t) \sim t^{-N(N-1)/2}.