Intermittent inverse-square Lévy walks are optimal for finding targets of all sizes

Lévy walks are random walk processes whose step lengths follow a long-tailed power-law distribution. Because of their abundance as movement patterns of biological organisms, substantial 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 Lévy walks are, in any manner, preferable strategies in higher dimensions than one. Here, we prove that in finite two-dimensional terrains, the inverse-square Lévy 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 Lévy walk fails to efficiently find either large targets or small ones. Our results shed new light on the Lévy foraging hypothesis and are thus expected to affect future experiments on animals performing Lévy walks.