Self-interacting random walks : aging, exploration and first-passage times

Abstract

Self-interacting random walks are endowed with long range memory effects that emerge from the interaction of the random walker at time t with the territory that it has visited at earlier times t < t. This class of non Markovian random walks has applications in a broad range of examples, ranging from insects to living cells, where a random walker modifies locally its environment -leaving behind footprints along its path, and in turn responds to its own footprints. Because of their inherent non Markovian nature, the exploration properties of self-interacting random walks have remained elusive. Here we show that long range memory effects can have deep consequences on the dynamics of generic self-interacting random walks ; they can induce aging and non trivial persistence and transience exponents, which we determine quantitatively, in both infinite and confined geometries. Based on this analysis, we quantify the search kinetics of self-interacting random walkers and show that the distribution of the first-passage time (FPT) to a target site in a confined domain takes universal scaling forms in the large domain size limit, which we characterize quantitatively. We argue that memory abilities induced by attractive self-interactions provide a decisive advantage for local space exploration, while repulsive self-interactions can significantly accelerate the global exploration of large domains.