Motivated by the \emph{L{\'e}vy flight foraging hypothesis} -- the premise that the movement of various animal species searching for food resembles a \emph{L{\'e}vy walk} -- we study the search efficiency of parallel L{\'e}vy walks on the infinite 2-dimensional grid. We assume that $k$ independent identical discrete-time L{\'e}vy walks, with exponent parameter $\alpha \in(1,+\infty)$, start simultaneously at the origin, and we are interested in the time $h_{\alpha,k,\ell}$ until some walk visits a given target node at distance $\ell$ from the origin. First, we observe that the total work, i.e., the product $k\cdot h_{\alpha,k,\ell}$, is at least $\Omega(\ell^2)$, for any combination of the parameters $\alpha,k,\ell$. Then we provide a comprehensive analysis of the time and work, for the complete range of these parameters. Our main finding is that for any $\alpha$, there is a specific choice of $k$ that achieves optimal work, $\tilde{\mathcal{O}}\left(\ell^2\right)$, whereas all other choices of $k$ result in sub-optimal work. In particular, in the interesting super-diffusive regime of $2 < \alpha < 3$, the optimal value for $k$ is $ \tilde \Theta\left(\ell^{1-(\alpha-2)}\right)$. Our results should be contrasted with several previous works showing that the exponent $\alpha = 2$ is optimal for a wide range of related search problems on the plane. On the contrary, in our setting of multiple walks which measures efficiency in terms of the natural notion of work, no single exponent is optimal: for each $\alpha$ (and $\ell$) there is a specific choice of $k$ that yields optimal efficiency.
翻译:受\ emph{L\\ e} vy friend for a propose} -- 各种动物物种寻找食物的移动类似于\ emph{L\\\\\ e} vy walk} -- 我们研究的是平行L\\ e} vy在无限的二维网格上行走的搜索效率。 我们假设, 独立独立的离散时间L\\ e} l\ e} vy 漫步, 以Exponent 参数$\ in (1, ⁇ ) 美元(美元) 开始同时从来源开始, 我们感兴趣的是时间 $(k) 美元, 直到有人步行访问某个目标, 从源头到美元。 首先, 我们观察的是总的工作效率, 即 $k\ cdoct halpha, k, k\ ell} 美元, 对于任何前一参数的组合来说, $( =xx 美元) 。 我们的主要结果是, 一个特定的工作结果是 $ 。