Distribution of first-passage times to specific targets on compactly explored fractal structures

Abstract

The distribution of the first passage times (FPT) of a one-dimensional random walker to a target site follows a power law F(t)~t(-3/2). We generalize this result to another situation pertinent to compact exploration and consider the FPT of a random walker with specific source and target points on an infinite fractal structure with spectral dimension d(s)<2. We show that the probability density of the first return to the origin has the form F(t)~t(d(s)/2-2), and the FPT to a specific target at distance r follows the law F(r,t)~r(d(w)-d(f)) t(d(s)/2-2), where d(w) and d(f) are the walk dimension and the fractal dimension of the structure, respectively. The distance dependence of F(r,t) reproduces the one of the mean FPT of a random walk in a confined domain.