Joint statistics of space and time exploration of $1d$ random walks

TL;DR

This paper derives the joint distribution of first-passage times and the number of visited sites for 1D random walks, providing a comprehensive understanding of the coupling between search kinetics and geometry.

Contribution

It introduces a general method to compute the joint distribution for 1D Markovian and non-Markovian processes, advancing the analysis of space exploration in random walks.

Findings

Explicit joint distribution expressions for several Markovian processes

A universal scaling form applicable to non-Markovian processes

Applications to trapping models and persistence properties

Abstract

The statistics of first-passage times of random walks to target sites has proved to play a key role in determining the kinetics of space exploration in various contexts. In parallel, the number of distinct sites visited by a random walker and related observables have been introduced to characterize the geometry of space exploration. Here, we address the question of the joint distribution of the first-passage time to a target and the number of distinct sites visited when the target is reached, which fully quantifies the coupling between kinetics and geometry of search trajectories. Focusing on 1-dimensional systems, we present a general method and derive explicit expressions of this joint distribution for several representative examples of Markovian search processes. In addition, we obtain a general scaling form, which holds also for non Markovian processes and captures the general dependence of the joint distribution on its space and time variables. We argue that the joint distribution has important applications to various problems, such as a conditional form of the Rosenstock trapping model, and the persistence properties of self-interacting random walks.