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From: callahan@cs.jhu.edu (Paul Callahan)
Newsgroups: sci.physics,sci.math,sci.electronics
Subject: Effective Resistance on an Infinite Mesh (Random Walk Question)
Message-ID: <callahan.667353668@newton.cs.jhu.edu>
Date: 23 Feb 91 23:34:45 GMT
Lines: 26

Is there a closed form solution for the effective resistance between two
points on an infinite 2-D mesh whose edges are unit resistors?  Ideally, I 
would like a formula in terms of x and y, the coordinates of one point, 
assuming the other is the origin.  I can see how to approximate the answer
by considering the effective resistance between two points on a continuous
infinite resistive sheet, so the resistance should grow roughly logarithmically
with the Euclidean distance between the points.  A related question (which 
might have a nicer closed form) would be the branch current for each resistor
in the mesh assuming two points are put at some fixed potential.  Again,
it would be useful to have a formula in terms of the coordinates.

A problem that turns out to be related (which is what I'm really interested in)
is the following.  Consider a random walk on an infinite mesh in which each step
consists of choosing one of the four incident horizontal and vertical edges with
equal probability.  Suppose the walk starts at the origin.  What is the expected
number of times that such a walk will return to the origin before arriving
at the point (x,y)?  Can this be expressed as a simple formula in terms of 
x and y?  

I've browsed through Doyle and Snell, _Random Walks and Electrical Networks_, 
which deals with this sort of thing, but I don't have access to a copy right 
now.  Pointers to any other appropriate references would be appreciated.  

--
Paul Callahan
callahan@cs.jhu.edu


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