Monte Carlo simulations for the site percolation problem are presented for lattices up to 64 x 106 sites. We investigate for the square lattice the variable-range percolation problem, where distinct trends with bond-length are found for the critical concentrations and for the critical exponentsß and?. We also investigate the layer problem for stacks of square lattices added to approach a simple cubic lattice, yielding critical concentrations as a functional of layer number as well as the correlation length exponent?. We also show that the exciton migration probability for a common type of ternary lattice system can be described by a cluster model and actually provides a cluster generating function.