bdm659@csc.anu.edu.au (03/03/91)
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
greg@garnet.berkeley.edu (Greg Kuperberg) (03/03/91)
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 potential for a current sink of one amp at a vertex of the mesh and a one amp current source "at infinity" and then using the superposition principle. As someone else also pointed out, you can immediately deduce that the e.r. between two adjacent nodes is 1/2, and as I showed in my previous post you can find all the effective resistances with more work. (In particular the e.r. between two diagonally separated nodes is 2/pi.) 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. What does it mean to have an infinite 2-D mesh whose edges are unit resistors? I decided that there are not one but two natural interpretations. On the one hand, you could have an infinite 2-D grid of disconnected electrical nodes, and you could proceed to wire in the resistors one by one. Each time you throw a resistor in the effective resistance between any two given nodes goes down, so the e.r. must necessarily settle to some limit. Clearly, the limit does not depend on the order that the resistors arrive. On the other hand, you could have an infinite 2-D grid of nodes all shorted together with superconducters, and you could replace the shorts one by one with unit resistors. In this case, the resistance goes up with each new resistor and therefore also must settle at a limit, and once again the order of the resistors doesn't matter as long as every resistor eventually arrives. The first interpretation is probably the one that most people have in mind when the hear the problem. However, the idea of placing a current source at infinity only jibes with the second interpretation. Thus, it is a crucial point that in 2 dimensions, the two answers that you get are the same. Roughly speaking, if the resistance between the periphery and the center is large, it doesn't matter if the periphery is insulated from itself or if it is shorted together. I haven't completely worked out a mathematical proof, but I'm confident that this reasoning validates my derivation of all of the effective resistances between two nodes in addition to supporting the claim that the e.r. between adjacent nodes is 1/2. It's a different story in 3 and higher dimensions, because there the resistance between the center and infinity for an infinite mesh is *not* infinite. Let us call the limit obtained by replacing insulation by resistors the upper limit of the e.r. and the limit obtained by replacing superconducters by resistors the lower limit of the e.r. I know that for a 3D infinite mesh the lower limit of the e.r. between two adjacent nodes is 1/3. But what is the upper limit? The best illustration of my point is the example of an infinite tree in which all vertices have valence 3 with a unit resistor running along each edge. In this case, the e.r. between two adjacent nodes is "obviously" 1, because the entire circuit is irrelevant except for the resistor connecting the two nodes directly, but it is equally "obvious" by symmetry and the superposition principle that the resistance is 2/3. How can this be? The symmetry argument implicitly assumes that the entire periphery of the tree is shorted together. Given that, the circuit reduces by Kirchoff's laws to a single resistor with resistance 2/3. ---- Greg Kuperberg Reply only to postings you like. greg@math.berkeley.edu Ignore postings you dislike.