Intermittent Inverse-Square Lévy Walks are Optimal for Finding Targets of All Sizes

Abstract: L\'evy walks are random walk processes whose step-lengths follow a long-tailed power-law distribution. Due to their abundance as movement patterns of biological organisms, significant theoretical efforts have been devoted to identifying the foraging circumstances that would make such patterns advantageous. However, despite extensive research, there is currently no mathematical proof indicating that L\'evy walks are, in any manner, preferable strategies in higher dimensions than one. Here we prove that in finite two-dimensional terrains, the inverse-square L\'evy walk strategy is extremely efficient at finding sparse targets of arbitrary size and shape. Moreover, this holds even under the weak model of intermittent detection. Conversely, any other intermittent L\'evy walk fails to efficiently find either large targets or small ones. Our results shed new light on the \emph{L\'evy foraging hypothesis}, and are thus expected to impact future experiments on animals performing L\'evy walks.