On 1/4/08, James Propp <jpropp@cs.uml.edu> wrote:
... Nobody's described the precise connection with random walks, so I'll bring this into the conversation: the effective resistance between (0,0) and (1,2) is equal to the reciprocal of 4p, where p is the probability that a random walker who starts at (0,0) will hit (1,2) before returning to (0,0). (With probability 1, the walker will eventually hit one or the other.) Here 4 arises as the sum of the reciprocals of the resistances on the edges incident to (0,0). If we used (1,1) in place of (1,2), the probability in question would be Pi/8. If you want to know why this equality is true, see Doyle and Snell's "Random Walks and Electric Networks", available at math.PR/0001057.
Yet another approach, which as Jim emphasises later comprises a raft of distinct modelling options [though people are better equipped to understnad equivalences between those than they seem to be in the resistor network case].
By the way, what Fred Lunnon calls "admittence", namely the reciprocal of resistance, is something I always thought was called "conductance". Is this ones of those transatlantic terminological differences? ...
Nah, just a spelling mistake. I have to admit not usually making any distinction between conductance = 1/resistance (real), and admittance = 1/impedance (complex), since I assume that anything done in the real domain carries over immediately to the complex. Thinking about this, though --- what happens in the original problem if we replace "1 ohm" by "1 microF" ?
... I can give more details (and maybe even try to do the simulation) if people are interested.
Presumably probabilistic simulations _do_ depend in an essential way on the resistances being real? It would be interesting to compare results. I limited myself to runs of the order of 5 minutes or so, using Maple on a Mac Powerbook --- longer runs might get another decimal place or two I suppose. Fred Lunnon