Complete visitation statistics of one-dimensional random walks

We develop a framework to determine the complete statistical behavior of a fundamental quantity in the theory of random walks, namely, the probability that n(1), n(2), n(3), ... distinct sites are visited at times t(1), t(2), t(3),... From this multiple-time distribution, we show that the visitation statistics of one-dimensional 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 to determine the visitation statistics of two important stochastic processes, the run-and-tumble particle and the biased random walk.