On the Search Efficiency of Parallel Lévy Walks on $\mathbf{Z}^2$

L\'evy walk is a popular movement model where an agent repeatedly samples a direction uniformly at random, and then travels in that direction for a distance which follows a power law distribution with exponent $\alpha\in (1,\infty)$. L\'evy walks and some of its famous variants, such as Brownian motion, have been subject to extensive empirical investigations in the continuous framework (i.e. continuous time and support space) since, essentially, they play an important role in theoretical biology: for instance, their search efficiency has been investigated in terms of the discovery rate of food, where the latter corresponds to points distributed according to a given density over $\mathbf{R}^2$. In that framework, it has been shown that the best efficiency is achieved by setting the parameter $\alpha$ to 2. Motivated by their importance, we provide the first rigorous and comprehensive analysis of the efficiency of L\'evy walks in the discrete setting, by estimating the search efficiency of k independent, parallel L\'evy walks starting from the origin. In more detail, the search efficiency of this parallel process is here described in terms of the hitting time with respect to (only) one target node of the 2-dimensional infinite grid, and the consequent total work, i.e., the total number of steps performed by all the agents. The study of distributed algorithms for k searching agents on an infinite grid that aim to minimize the hitting time of a target node has been considered in Distributed Computing under the name of ANTS Problem (Feinerman et al. PODC 2012). Our probabilistic analysis of L\'evy walks implies the following main novel contributions: I. We show that L{\'e}vy walks provide a biologically well-motivated, time-invariant search protocol for the ANTS Problem which does not require any communication among the k agents, and which improves the state of the art in terms of efficiency, for a wide range of the parameter k. II. In contrast with the apparent general optimality of the setting $\alpha = 2$, suggested by the discovery rate mentioned above, we show that the best exponent with respect to a natural measure, such as the total work of the process, directly depends on the number of searching agents k.