Invariance relations for random walks on square-planar lattices

We outline a systematic procedure for obtaining exact analytic expressions for the mean walklength 〈n〉 for a random walker transiting on a d = 2 square-planar lattice with a single deep trap subject to periodic boundary conditions. As demonstrated in our earlier work on hexagonal lattices [Chem. Phys. Lett. 371 (2003) 365], the procedure depends on generalizing Montroll’s first invariance relation ∑(n = 1) to nth order. The central result of this work is the following exact analytic expression for ∑(n) for square-planar lattices: ) = 4 ∑ i = 0 n - 1 3 i N - 1 3 ! 5 · 3 n + 12 ( n - 2 ) 3 n - 1 + 3 .