Random walks with ‘spontaneous emission’ on lattices with periodically distributed imperfect traps

We study random walks on d-dimensional lattices with periodically distributed traps in which the walker has a finite probability per step of disappearing from the lattice and a finite probability of escaping from a trap. General expressions are derived for the total probability that the walk ends in a trap and for the moments of the number of steps made before this happens if it does happen. The analysis is extended to lattices with more types of traps and to a model where the trapping occurs during special steps. Finally, the Green's function at the origin G(0; z) for a finite lattice with periodic boundary conditions, which enters into the main expressions, is studied more closely. A generalization of an expression for G(0; 1) for the square lattice given by Montroll to values of z different from, but close to, 1 is derived.