Resistance between any 2 nodes on an infinite square grid

This question is motivated by this xkcd comic strip .

The problem is indeed interesting, and my first recollection upon reading this was a similar problem in the book Problems in General Physics by I.E.Irodov(which, in my humble opinion,is a masterpiece).

The question I wanted to ask is, are there any generalizations of the problem known? Can be find the resistance between any two nodes of the grid as a function of the distance between the nodes?

Many thanks in advance!

• possible duplicate of On this infinite grid of resistors, what's the equivalent resistance? – Mark Eichenlaub May 24 '11 at 6:42
• @mark: This question is regarding the resistance between any two nodes. – Koundinya Vajjha May 24 '11 at 7:13
• I must note that Irodov solution (using symmetry and superposition) works only for two adjacent nodes. – Kostya May 24 '11 at 8:54
• @Kound It is functionally the same question. If you read the answers to the previous question in detail, you would have known the answer to this one. In fact, both questions were answered with a link to exactly the same web page. – Mark Eichenlaub May 24 '11 at 11:28
• There are more generalizations, how about 3D cubic grid? n-cubic grid? – TROLLHUNTER May 24 '11 at 12:02

Yes, it is possible. For example Kevin Brown did here and here including this table. so for the xkcd problem the answer is $-\frac{1}{2}+\frac{4}{\pi} \approx 0.773$.

• Who is this Kevin Brown? I bought his book and I don't know anything about him. – Kasper Meerts May 24 '11 at 12:42
• @kmm: He is deliberately inconspicuous. In MathPages, his name appears on pictures of the covers of his books and in one other place. – Henry May 24 '11 at 13:14
• That's strange. I'd just want to e-mail him saying I immensely enjoyed his book. – Kasper Meerts May 24 '11 at 20:17

As far as I know, the first solution to the general problem is given by Cserti,

József Cserti. Application of the lattice Green’s function for calculating the resistance of an infinite network of resistors. Am. J. Phys. 68 no. 10, pp. 896 (2000). doi:10.1119/1.1285881, arXiv:cond-mat/9909120 [cond-mat.mes-hall])

using lattice Green's functions (and there are references to previous partial solutions). For your first question, the recurrence relation that gives the resistance between nodes in a square lattice is equation 32. The paper also describes how to solve or derive asymptotics for rectangular lattices, triangular lattices, honeycombs, and cubic lattices in higher dimensions.