Year: 2004
Type: article
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 by Fisher (1984 J. Stat. Phys. 34 667) and Krattenthaler et al (2000 J. Phys. A: Math. Gen. 33 8835) 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: .
Authors: Alan J Bray, Karen Winkler
Institution: University of Manchester
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
OpenAlex ID: W2154851143