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

TL;DR

This paper proves that Cauchy Le9vy walks are uniquely optimal for 3D search tasks, achieving scale-invariant detection across various target shapes and sizes, with implications for biological and engineered search systems.

Contribution

It provides a rigorous mathematical proof that the Cauchy Le9vy walk strategy is uniquely optimal for 3D search across diverse target shapes and sizes, establishing a foundation for the Le9vy flight foraging hypothesis.

Findings

Cauchy strategy (b5=2) achieves near-optimal, scale-invariant detection in 3D.

Detection time scales as volume divided by surface area for convex targets.

Shape and elongation significantly influence search efficiency in 3D.

Abstract

Target shape, not just size, plays a pivotal role in determining detectability during random search. We analyze intermittent L\'evy walks in three dimensions, and mathematically prove that the widely observed Cauchy strategy (L\'evy exponent $\mu = 2$) uniquely achieves scale-invariant, near-optimal detection across a broad spectrum of target sizes and shapes. In a domain of volume $n$ with boundary conditions, expected detection time for a convex target of surface area $\Delta$ optimally scales as $n/\Delta$. Conversely, L\'evy strategies with $\mu < 2$ are slow at detecting targets with large surface area-to-volume ratios, while those with $\mu > 2$ excel at finding large elongated shapes but degrade as targets become wider. Our results further indicate a continuous geometric transition: volume dictates detection near $\mu = 1$, ceding dominance to surface area as $\mu \to 2$, after which surface area and elongation couple to govern detection. Ultimately, 3D search introduces a pronounced sensitivity to target shape that is absent in lower dimensions. Our work provides a rigorous foundation for the L\'evy flight foraging hypothesis in 3D by establishing the scale-invariant optimality of the Cauchy walk. Furthermore, our results reveal dimensionality-driven shape vulnerabilities and offer testable predictions for biological and engineered systems.