# Random walks on resistive network

I have been referring to a paper http://arxiv.org/abs/physics/0405135 to determine the effective resistance using random walks for an infinite square resistive lattice

Though the author seems to indicate this as a simple problem (maybe i am missing something) i have been unable to prove this

$∆_{AB}$ = $\frac{1}{2p_{AB}}$

where,

$∆_{AB}$ = $\sum_{n=0}^\infty$ $(P_{n}(A) − P{n}(B))$

$P_{n}(x):$Probability that a Random walker after n steps is found at x

$p_{AB}:$Probability that a random walker, starting at A, gets to B before returning to A