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Self-interacting random walks are endowed with long range memory effects thatemerge from the interaction of the random walker at time $t$ with the territorythat it has visited at earlier times $t'<t$. This class of non Markovian randomwalks has applications in a broad range of examples, ranging from insects toliving cells, where a random walker modifies locally its environment -- leavingbehind footprints along its path, and in turn responds to its own footprints.Because of their inherent non Markovian nature, the exploration properties ofself-interacting random walks have remained elusive. Here we show that longrange memory effects can have deep consequences on the dynamics of genericself-interacting random walks ; they can induce aging and non trivialpersistence and transience exponents, which we determine quantitatively, inboth infinite and confined geometries. Based on this analysis, we quantify thesearch kinetics of self-interacting random walkers and show that thedistribution of the first-passage time (FPT) to a target site in a confineddomain takes universal scaling forms in the large domain size limit, which wecharacterize quantitatively. We argue that memory abilities induced byattractive self-interactions provide a decisive advantage for local spaceexploration, while repulsive self-interactions can significantly accelerate theglobal exploration of large domains.

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