On Measures of Average Degree for Lattices

The usual definition of average degree for a non-regular lattice has the disadvantage that it takes the same value for many lattices with clearly different connectivity. We introduce an alternative definition of average degree, which better separates different lattices. These measures are compared on a class of lattices and are analyzed using a Markov chain describing a random walk on the lattice. Using the new measure, we conjecture the order of both the critical probabilities for bond percolation and the connective constants for self-avoiding walks on these lattices.