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Motivated by the emph{L{e}vy foraging hypothesis} -- the premise that various animal species have adapted to follow emph{L{e}vy walks} to optimize their search efficiency -- we study the parallel hitting time of L{e}vy walks on the infinite two-dimensional grid.We consider $k$ independent discrete-time L{e}vy walks, with the same exponent $alpha in(1,infty)$, that start from the same node, and analyze the number of steps until the first walk visits a given target at distance $ell$.We show that for any choice of $k$ and $ell$ from a large range, there is a unique optimal exponent $alpha_{k,ell} in (2,3)$, for which the hitting time is $tilde O(ell^2/k)$ w.h.p., while modifying the exponent by an $epsilon$ term increases the hitting time by a polynomial factor, or the walks fail to hit the target almost surely.Based on that, we propose a surprisingly simple and effective parallel search strategy, for the setting where $k$ and $ell$ are unknown:The exponent of each L{e}vy walk is just chosen independently and uniformly at random from the interval $(2,3)$.This strategy achieves optimal search time (modulo polylogarithmic factors) among all possible algorithms (even centralized ones that know $k$).Our results should be contrasted with a line of previous work showing that the exponent $alpha = 2$ is optimal for various search problems.In our setting of $k$ parallel walks, we show that the optimal exponent depends on $k$ and $ell$, and that randomizing the choice of the exponents works simultaneously for all $k$ and $ell$.
Intermittent stochastic processes appear in a wide field, such as chemistry, biology, ecology, and computer science. This paper builds up the theory of intermittent continuous time random walk (CTRW) and L{e}vy walk, in which the particles are stocha
Recent experiments (G. Ariel, et al., Nature Comm. 6, 8396 (2015)) revealed an intriguing behavior of swarming bacteria: they fundamentally change their collective motion from simple diffusion into a superdiffusive L{e}vy walk dynamics. We introduce
The estimation of functions with varying degrees of smoothness is a challenging problem in the nonparametric function estimation. In this paper, we propose the LABS (L{e}vy Adaptive B-Spline regression) model, an extension of the LARK models, for the
The Wiener-Hopf factorization is obtained in closed form for a phase type approximation to the CGMY L{e}vy process. This allows, for the approximation, exact computation of first passage times to barrier levels via Laplace transform inversion. Calibr
In a step reinforced random walk, at each integer time and with a fixed probability p $in$ (0, 1), the walker repeats one of his previous steps chosen uniformly at random, and with complementary probability 1 -- p, the walker makes an independent new