# 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

(For more detailed description refer to the link)

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One finds a full explanation of this formula and much more about the connection between random walks and electric networks in the little gem Random walks and electric networks by Peter G. Doyle and J. Laurie Snell, freely available.

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I think that link-only answers are frowned on here. Could you condense the explanation for us? –  rob May 8 at 4:43