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I've been exposed to various problems involving infinite circuits but never seen an extensive treatment on the subject. The main problem I am referring to is

Given a lattice L, we turn it into a circuit by placing a unit resistance in each edge. We would like to calculate the effective resistance between two points in the lattice (Or an asymptotic value for when the distance between the points gets large).

I know of an approach to solve the above introduced by Venezian, it involves superposition of potentials. An other approach I've heard of, involves lattice Green functions (I would like to read more about this). My first request is for a survey/article that treats these kind of problems (for the lattices $\mathbb{Z}^n$, Honeycomb, triangular etc.) and lists the main approaches/results in the field.

My second question (that is hopefully answered by the request above) is the following:

I noticed similarities in the transition probabilities of a Loop-erased random walk and the above mentioned effective resistances in $\mathbb{Z}^2$. Is there an actual relation between the two? (I apologize if this is obvious.)

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3 Answers

up vote 3 down vote accepted

The book by Peres and Lyons, freely available here http://php.indiana.edu/~rdlyons/prbtree/prbtree.html, should give you much information at least for the probability part of the question.

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Thanks! It looks like a great reference. –  Gjergji Zaimi Sep 2 '10 at 9:06
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If you are still interested in this, you may want to have a look in Section 6 of http://www.sciencedirect.com/science/article/pii/0095895690900658 by Thomassen. He proves for example that the effective resistance between adjacent vertices of $Z^2$ is 1/2. I don't think there is mention to LERW though.

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A Google Scholar search for "random walks and electrical networks" will bring up a text by Doyle and Snell that is now available online; for additional references check the citations.

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Thanks for the reference, I have browsed through that monograph before. I think the problem I discuss above wasn't solved at the time Doyle and Snell published their monograph, also they don't treat any relations with LERW. –  Gjergji Zaimi Feb 1 '10 at 11:02
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You may find Pemantle's article in the cites interesting: arxiv.org/abs/math/0404043 –  Steve Huntsman Feb 1 '10 at 11:09
    
I'll look into it, thanks. –  Gjergji Zaimi Feb 1 '10 at 12:23
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