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



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> 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). 
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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.)



Infinite electrical networks and possible connections with LERW
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.


