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 $n_1$, $n_2$, $n_3$, . . . distinct sites are visited at times $t_1$, $t_2$, $t_3$, ... . From this multiple-time distribution, we show that the visitation statistics of 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.

Main text: 5 pages, 4 figures; Supplementary material: 12 pages, 3 figures

References in corpus (3)

• Sedimentation, trapping, and rectification of dilute bacteria
• Collective dynamics of social annotation
• Joint distributions of partial and global maxima of a Brownian Bridge

Cited by in corpus (6)

• Universal exploration dynamics of random walks
• Record Ages of Scale Invariant non-Markovian Random Walks
• Range-controlled random walks
• From Maximum of Intervisit Times to Starving Random Walks
• The joint distribution of first return times and of the number of distinct sites visited by a 1D random walk before returning to the origin
• Full Record Statistics of 1d Random Walks