Exact Algorithm ford-Dimensional Walks on Finite and Infinite Lattices with Traps

An exact algorithm is formulated to calculate the expected walk length $〈n〉$ for a walker (atom, molecule) undergoing random displacements on a finite or infinite (periodic) $d$-dimensional lattice with traps (reactive sites). The method is illustrated for the case of a single deep trap surrounded by shallow traps and the calculated value of $〈n〉$ agrees to within 0.3% of the Monte Carlo result for all lattices considered. The theory introduced is capable of generalization to many new classes of problems in lattice statistics.