A conditionally sure ergodic theorem with an application to percolation

We prove an apparently new type of ergodic theorem, and apply it to the site percolation problem on sparse random sublattices of ZdZd (d≥2d≥2), called “lattices with large holes”. We show that for every such lattice the critical probability lies strictly between zero and one, and the number of the infinite clusters is at most two with probability one. Moreover for almost every such lattice, the infinite cluster, if it exists, is unique with probability one.