First-passage times in multi-scale random walks: the impact of movement scales on search efficiency

TL;DR

This paper analytically investigates how multi-scale random walks influence search efficiency in a finite domain, revealing that combining two movement scales optimally can outperform traditional strategies, with implications for understanding search behaviors.

Contribution

It provides exact expressions for mean-first passage times in multi-scale random walks and demonstrates that two well-chosen scales suffice for optimal search in asymmetric regimes.

Findings

Two-scale strategies outperform ballistic and Lévy strategies.

Optimal search requires prior information to adjust movement scales.

Adding more than two scales does not improve efficiency.

Abstract

An efficient searcher needs to balance properly the tradeoff between the exploration of new spatial areas and the exploitation of nearby resources, an idea which is at the core of scale-free L\'evy search strategies. Here we study multi-scale random walks as an approximation to the scale- free case and derive the exact expressions for their mean-first passage times in a one-dimensional finite domain. This allows us to provide a complete analytical description of the dynamics driving the asymmetric regime, in which both nearby and faraway targets are available to the searcher. For this regime, we prove that the combination of only two movement scales can be enough to outperform both balistic and L\'evy strategies. This two-scale strategy involves an optimal discrimination between the nearby and faraway targets, which is only possible by adjusting the range of values of the two movement scales to the typical distances between encounters. So, this optimization necessarily requires some prior information (albeit crude) about targets distances or distributions. Furthermore, we found that the incorporation of additional (three, four, ...) movement scales and its adjustment to target distances does not improve further the search efficiency. This allows us to claim that optimal random search strategies in the asymmetric regime actually arise through the informed combination of only two walk scales (related to the exploitative and the explorative scale, respectively), expanding on the well-known result that optimal strategies in strictly uninformed scenarios are achieved through L\'evy paths (or, equivalently, through a hierarchical combination of multiple scales).