Effect of the search space dimensionality for finding close and faraway targets in random searches.

Abstract

We investigate the dependence on the search space dimension of statistical properties of random searches with Lévy α-stable and power-law distributions of step lengths. We find that the probabilities to return to the last target found (P_{0}) and to encounter faraway targets (P_{L}), as well as the associated Shannon entropy S, behave as a function of α quite differently in one (1D) and two (2D) dimensions, a somewhat surprising result not reported until now. While in 1D one always has P_{0}≥P_{L}, an interesting crossover takes place in 2D that separates the search regimes with P_{0}>P_{L} for higher α and P_{0}<P_{L} for lower α, depending on the initial distance to the last target found. We also obtain in 2D a maximum in the entropy S for α∈(0,2], not observed in 1D apart from the trivial α→0 ballistic limit. Improving the understanding of the role of dimensionality in random searches is relevant in diverse contexts, as in the problem of encounter rates in biology and ecology.