On the Search Efficiency of Parallel L{é}vy Walks on ${\mathbb Z}^2$

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.