Intermittent Cauchy walks enable optimal 3D search across target shapes and sizes

Abstract

Efficiently locating rare targets requires balancing the exploration of novel space with the exploitation of local vicinities. While this tradeoff is well-documented in simplified environments, its behavior in realistic three-dimensional environments remains unclear. Here we analyze intermittent Lévy walks in three dimensions and show that search efficiency becomes strongly dependent on target geometry, with volume, surface area, and elongation governing performance across the Lévy spectrum. Against this geometric complexity, the Cauchy walk ($\mu =2$) emerges as a robust, near-optimal solution across a wide range of target geometries. Specifically, we prove that for convex targets of surface area $\Delta$ in a domain of volume $n$, the optimal expected detection time scales as $n/\Delta$, and that the Cauchy walk achieves this scaling up to polylogarithmic factors, while other Lévy strategies are suboptimal in different geometric regimes. Our findings provide a theoretical framework for biological movement patterns, including marine predator foraging and immune surveillance, and inform the design of efficient search algorithms for engineered systems such as autonomous nanoscale explorers.

Accepted for publication in Proceedings of the National Academy of Sciences (PNAS)