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

2026-03-11

AI Summary

This study addresses the challenge of target geometry significantly influencing search efficiency in three-dimensional random exploration, where conventional strategies lack universality. By integrating an intermittent Lévy walk model with probabilistic analysis and geometric measure theory, the work rigorously establishes—for the first time—that Cauchy walks (Lévy exponent μ = 2) achieve near-optimal and scale-invariant detection performance for any convex target in three dimensions. The research uncovers a geometric transition mechanism underlying shape sensitivity in high-dimensional spaces and derives the optimal scaling law for mean detection time as n/Δ, where n denotes target volume and Δ its surface area. These findings provide testable theoretical predictions with direct implications for biological foraging behaviors and engineered search systems.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Search
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Web Mining and Content Analysis: Models for Web evolution
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

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.

Problem

• 3D search
• target shape
• detectability
• scale-invariance
• Lévy walks

Innovation

• Cauchy walk
• Lévy flight
• 3D search
• scale-invariant optimality
• target shape sensitivity