The Search Efficiency of Intermittent Lévy walks Optimally Scales with Target Size
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We address the general question of what is the best strategy to employ when searching for targets of unpredictable sizes. Motivated by biological scenarios, we focus on generalized random walk strategies, in which step-lengths follow a given distribution, while their direction is randomly uniform. To capture common biological scenarios in which motion degrades detection, we further restrict attention to intermittent settings, which means that the searcher can detect a target only while being "immobile", that is, in-between moving steps. We prove that in a finite two-dimensional torus, an intermittent Lévy walk whose step-lengths are distributed according to an inverse-square power-law is a near-optimal strategy for finding sparse targets of any size and shape. Specifically, in a square torus of area n, and assuming that the detection radius is normalized to 1, the strategy finds any connected set of diameter D in Õ(n/D) expected time, whereas Ω(n/D) is an unconditional lower bound on the expected time, that holds even when assuming that the shape and size of the target are known. Furthermore, this particular Lévy process stands in stark contrast to many other basic intermittent processes, including all other Lévy walks, which we prove to be extremely inefficient for wide ranges of target scales. Our results thus provide strong theoretical support for the optimality and robustness of intermittent Lévy walks under general conditions.
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