We consider a random stationary vector field on a multidimensional lattice and investigate flow-connected subsets of the lattice invariant under the action of the associated flow. The subsets of primary interest are cycles, and vortices each of which is the set of orbits terminating in the same cycle. We prove that with probability 1 each vortex only involves a finite number of sites of the lattice. Under the assumption of independence of the vector field in different sites, we find that with probability 1 the vortices exhaust all possible maximal flow-connected invariant subsets of the lattice if and only if the probability of existence of a cycle is positive. Thus, if cycles exist, a particle under the action of the flow only moves within a bounded region, i.e., it is completely localized.