Complete Visitation Statistics of 1d Random Walks

TL;DR

This paper develops a comprehensive framework for analyzing the complete visitation statistics of one-dimensional random walks, revealing their temporal correlations and non-Markovian behavior, with applications to trapping problems and specific stochastic processes.

Contribution

It introduces a novel framework to compute full multi-time visitation distributions in 1d random walks, uncovering their non-Markovian properties and applying these insights to various stochastic models.

Findings

Visitation statistics are temporally correlated.

Random walks exhibit non-Markovian behavior.

Derived results for trapping problems and specific processes.

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.