Intermittent Cauchy walks enable optimal 3D search across target shapes and sizes
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| Format: | Preprint |
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2026
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| author | Stromieri, Matteo Natale, Emanuele Korman, Amos |
| author_facet | Stromieri, Matteo Natale, Emanuele Korman, Amos |
| contents | Target shape, not just size, plays a pivotal role in determining detectability during random search. We analyze intermittent Lévy walks in three dimensions, and mathematically prove that the widely observed Cauchy strategy (Lévy exponent $μ= 2$) 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 $Δ$ optimally scales as $n/Δ$. Conversely, Lévy strategies with $μ< 2$ are slow at detecting targets with large surface area-to-volume ratios, while those with $μ> 2$ excel at finding large elongated shapes but degrade as targets become wider. Our results further indicate a continuous geometric transition: volume dictates detection near $μ= 1$, ceding dominance to surface area as $μ\to 2$, 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 Lévy 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_10655 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Intermittent Cauchy walks enable optimal 3D search across target shapes and sizes Stromieri, Matteo Natale, Emanuele Korman, Amos Data Structures and Algorithms Probability Target shape, not just size, plays a pivotal role in determining detectability during random search. We analyze intermittent Lévy walks in three dimensions, and mathematically prove that the widely observed Cauchy strategy (Lévy exponent $μ= 2$) 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 $Δ$ optimally scales as $n/Δ$. Conversely, Lévy strategies with $μ< 2$ are slow at detecting targets with large surface area-to-volume ratios, while those with $μ> 2$ excel at finding large elongated shapes but degrade as targets become wider. Our results further indicate a continuous geometric transition: volume dictates detection near $μ= 1$, ceding dominance to surface area as $μ\to 2$, 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 Lévy 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. |
| title | Intermittent Cauchy walks enable optimal 3D search across target shapes and sizes |
| topic | Data Structures and Algorithms Probability |
| url | https://arxiv.org/abs/2603.10655 |