Vicious walkers, flows and directed percolation

D.K. Arrowsmith, P. Mason and J.W. Essam

Physica A: Statistical Mechanics and its Applications, 1991, vol. 177, issue 1, 267-272

Abstract: It is shown that the problem of m vicious random walkers is equivalent to the enumeration of integer natural flows on a directed square lattice with maximum flow m. An explicit formula for the number of such flows is given as a polynomial in m and is shown to have the same asymptotic form as Fisher's determinantal result. The expected number of such flows on directed bond or site percolation clusters is also a polynomial in m which reduces to the pair connectedness when m = 0. Consequently directed percolation may be seen as a problem of interacting vicious random walkers. Exact results for m = 1 and 2 are given.

Date: 1991
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Persistent link: https://EconPapers.repec.org/RePEc:eee:phsmap:v:177:y:1991:i:1:p:267-272

DOI: 10.1016/0378-4371(91)90163-7

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