Vicious Walkers in a Potential

Details Vicious Walkers in a Potential Vicious Walkers in a Potential

2004-03-08

arxiv.org/abs/cond-mat/0403220v1

J. Phys. A 37, 5493 (2004)

Physics

Condensed Matter

Statistical Mechanics

5 pages

Scientific paper

10.1088/0305-4470/37/21/001

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}.

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