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

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.

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• First-passage functionals of Brownian motion in logarithmic potentials and heterogeneous diffusion
• Extreme Value Statistics and Arcsine Laws of Brownian Motion in the Presence of a Permeable Barrier
• Extremal statistics for a one-dimensional Brownian motion with a reflective boundary
• First-passage and extreme value statistics for overdamped Brownian motion in a linear potential
• 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
• Persistence exponents of self-interacting random walks
• Full Record Statistics of 1d Random Walks
• First-passage processes in a deterministic one-dimensional cellular automaton model of traffic flow