L\'evy walks are random walks processes whose step length follows a long-tailed power law distribution. Due to their abundance as movement patterns of URLanisms, significant theoretical efforts have been devoted to identify the foraging circumstances that would make such patterns advantageous Viswanathan et al. Nature, 1999 . Despite numerous attempts , there is currently no conclusive analytical evidence that L\'evy flights are preferable strategies in higher dimensions than one. Here we show that the optimality of inverse-square L\'evy walks in two-dimensions becomes striking when targets are sparse and unpredictable in size, and when detection is weak. Specifically, we prove that under the intermittent model, in which detection is not possible while moving ballistically B\'enichou et al. Reviews of Modern Physics, 2011%DIFDELCMD < ]%%% , this strategy optimally finds sparse targets of any size and shape. That is, 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%DIFDELCMD < \tilde{O}%%% (n/D) expected time, whereas \Omega(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 L\'evy process stands in stark contrast to many other basic intermittent processes, including all other L\'evy walks, whose competitive ratio is shown to be polynomial in n, for wide ranges of target scales . Our results thus provide strong theoretical support for the optimality and robustness of intermittent\emph{  L\'evy walks under general conditions .
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 URLanisms, 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 %DIFDELCMD < ]%%% walk strategy is extremely efficient at finding sparse targets of arbitrary size and shape. %DIFDELCMD < \tilde{O}%%% 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 .
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