Optimization of random searches on regular lattices

We investigate random searches on isotropic and topologically regular square and triangular lattices with periodic boundary conditions and study the efficiency of search strategies based on a power-law distribution P(l)similar to l(-mu) of step lengths l. We consider both destructive searches, in which a target can be visited only once, and nondestructive searches, when a target site is always available for future visits. We discuss (i) the dependence of the search efficiency on the choice of the lattice topology, (ii) the relevance of the periodic boundary conditions, (iii) the behavior of the optimal power-law exponent mu(opt) as a function of target site density, (iv) the differences between destructive and nondestructive environments, and finally (v) how the results for the discrete searches differ from the continuous cases previously studied.