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 hitting time of parallel L{\'e}vy walks on the infinite 2-dimensional grid. L{\'e}vy walks are characterized by a parameter $\alpha \in(1,+\infty)$, that is the exponent of the power law distribution of the time intervals at which the moving agent randomly changes direction. In the setting we consider, called the ANTS problem (Feinerman et al. PODC 2012), $k$ independent discrete-time L{\'e}vy walks start simultaneously at the origin, and we are interested in the time $h\_{k,\ell}$ before some walk visits a given target node on the grid, at distance $\ell$ from the origin. In this setting, we provide a comprehensive analysis of the efficiency of L{\'e}vy walks for the complete range of the exponent $\alpha$. For any choice of $\alpha$, we observe that the \emph{total work} until the target is visited, i.e., the product $k \cdot h\_{k,\ell}$, is at least $\Omega(\ell^2)$ with constant probability. Our main result is that the right choice for $\alpha$ to get optimal work varies between $1$ and $3$, as a function of the number $k$ of available agents. For instance, when $k = \tilde \Theta(\ell^{1-\epsilon})$, for some positive constant $\epsilon < 1$, then the unique optimal setting for $\alpha$ lies in the \emph{super-diffusive} regime $(2,3)$, namely, $\alpha = 2+\epsilon$. Our results should be contrasted with various previous works in the continuous time-space setting showing that the exponent $\alpha = 2$ is optimal for a wide range of related search problems on the plane.