Optimizing random searches on three-dimensional lattices

Search is a universal behavior related to many types of intelligent individuals. While most studies have focused on search in two or infinite-dimensional space, it is still missing how search can be optimized in three-dimensional space. Here we study random searches on three-dimensional (3d) square lattices with periodic boundary conditions, and explore the optimal search strategy with a power-law step length distribution, p(l)∼l^−μ, known as Lévy flights. We find that compared to random searches on two-dimensional (2d) lattices, the optimal exponent μ_opt on 3d lattices is relatively smaller in non-destructive case and remains similar in destructive case. We also find μ_opt decreases as the lattice length in z direction increases under high target density. Our findings may help us to understand the role of spatial dimension in search behaviors.