🚀 go-pugleaf

In article <1991Mar2.230545.21870@agate.berkeley.edu>, greg@garnet.berkeley.edu (Greg Kuperberg) writes:
> In article <callahan.667353668@newton.cs.jhu.edu> callahan@cs.jhu.edu (Paul Callahan) writes:
>>Is there a closed form solution for the effective resistance between two
>>points on an infinite 2-D mesh whose edges are unit resistors?
>
> I posted a solution to this problem which consisted of finding the
>[...]
>
> In that post I glibly wrote off the serious issue which Brendan McKay
> brought up of "unphysical", i.e. mathematically fallacious, arguments:
>
> In article <1991Feb28.203048.13891@agate.berkeley.edu> greg@math.berkeley.edu writes:
>>As Brendan McKay pointed out, the voltages for the solution are unbounded
>>as you go to infinity.  But it's a well-posed mathematical problem
>>anyway.
>
> I thought about this issue some more last night and came to a strange
> conclusion:  The argument using the current source at infinity is valid
> precisely because of and not in spite of the fact that the resistance
> between a vertex and the periphery is infinite.

Excellent.  Carsten Thomassen often made this peculiar observation
in lectures he gave on the subject some years ago.

Here is an extract from an item I posted on 15 Dec 1989.  I don't
remember if anybody located journal references for the papers I mention.

=========

Carsten Thomassen, Mathematical Institute, The Technical University of Denmark,
Building 303, DK-2800 Lyngby, Denmark, has written a number of papers on this
subject.  He doesn't use e-mail, unfortunately.  Titles include "Resistances
and currents in infinite electrical networks" and "Transient random walks,
harmonic functions, and electrical currents in infinite resistive networks".
I only have these in preprint form, but they should be in print by now.
Try looking up Math. Reviews, or Science Citation Index.

Amongst the theorems proved by Thomassen is the following:

    If an infinite connected edge-transitive network of moderate growth
    is regular of degree d, then the resistance between a pair of adjacent
    nodes is 2/d.

Definitions:
   edge-transitive = there is a symmetry taking any edge onto any other edge
   moderate growth = the node set can be partitioned into a sequence of
                     disjoint finite classes V[0], V[1], ...  such that
                 (a) there is no edge between V[i] and V[j] if i and j differ
                     by more than one,
                 (b) |V[k]| / (|V[0]| + ... + |V[k]|) -> 0 as k -> infinity,
                     where |.| denotes cardinality.
                 [Example: Take V[k] to be the set of nodes at distance
                  k from some fixed node.  An example of moderate growth is
                  when |V[k]| is roughly polynomial in growth rate, as it
                  will be for many nice networks in finitely many dimensions.]

======

> Greg Kuperberg                 Reply only to postings you like.
> greg@math.berkeley.edu         Ignore postings you dislike.

Brendan McKay