Vicious walkers and directed polymer networks in general dimensions

Abstract

A number, p, of vicious random walkers on a D-dimensional lattice is considered. ‘‘Vicious walkers’’ describes the situation when two or more walkers arrive at the same lattice site and annihilate one another, and consequently their walks terminate. In certain cases the generating function R(u)[S(u)] for the number of configurations ${\mathit{R}}_{\mathit{s}}$[${\mathit{S}}_{\mathit{s}}$] of walkers which reunite [survive] after s steps is expressed in terms of generalized hypergeometric functions. The critical exponents associated with these functions are in agreement with known results for Brownian paths. The critical dimension D=2 also agrees with that found for the continuum limit, and logarithmic corrections are discussed. Vicious walker configurations correspond to directed polymer networks in d=D+1 dimensions, and in the case D=1 they also correspond to directed integer flows in which the flow in any bond is in the range 0 to p. (c) 1995 The American Physical Society