Complete Visitation Statistics of 1d Random Walks

Abstract

We develop a framework to determine the complete statistical behavior of a fundamental quantity in the theory of random walks, namely, the probability that n1, n2, n3, . . . distinct sites are visited at times t1, t2, t3, . . .. From this multiple-time distribution, we show that the visitation statistics of 1d random walks are temporally correlated and we quantify the non-Markovian nature of the process. We exploit these ideas to derive unexpected results for the two-time trapping problem and also to determine the visitation statistics of two important stochastic processes, the run-and-tumble particle and the biased random walk.