Directed paths on percolation clusters

Abstract

We consider a variant of the problem of directed polymers on a disordered lattice, in which the disorder is “geometrical” in nature. In particular, we allow a finite probability for each bond to be absent from the lattice. We show, through the use of numerical and scaling arguments on both Euclidean and hierarchical lattices, that the model has two distinct scaling behaviors, depending upon whether the concentration of bonds on the lattice is at or above the directed percolation threshold. We are particularly interested in the exponentsω andζ, defined by δf∼tω and δx∼tζ, describing the free-energy and transverse fluctuations, respectively. Above the percolation threshold, the scaling behavior is governed by the standard “random energy” exponents (ω=1/3 and ζ=2/3 in 1+1 dimensions). At the percolation threshold, we predict (and verify numerically in 1+1 dimensions) the exponentsω=1/2 and ζ=v⊥/v∥, where v⊥ and v∥ are the directed percolation exponents. In addition, we predict the absence of a “free phase” in any dimension at the percolation threshold.