Random walks on pseudo-lattices

The problem of a Pólya (unbiased, nearest-neighbor stepping) random walk is solved for several pseudo-lattices related to the Bethe lattice (Cayley tree). In each case, the solution is derived by projecting the walk on a pseudo-lattice onto a walk on a linear chain with internal states and a point defect. Random walk statistics exhibited explicitly include the probability of return to the starting point, the mean time to return if return does occur, and the asymptotic behavior of the expected number of distinct sites visited in a walk of long duration. Conjectured relationships between random walk statistics and percolation theory are discussed in the context of pseudo-lattices.