© 1984 by London Mathematical Society
RANDOM ELECTRICAL NETWORKS ON COMPLETE GRAPHS
Schools of Mathematics, University of Bristol, University Walk Bristol BS8 1TW
Mathematics Department Cornell University Ithaca, N.Y. 14853, U.S.A.
Received 22 August 1983.
A random electrical network is a graph G with edges which are electrical connections, whose resistances form a family of independent identically distributed random variables. We consider the case when G is a complete graph on n + 2 vertices and a typical edge-resistance R has distribution
 |
where 0 
(n) n and F is a fixed distribution function concentrated on [0, 
). It turns out that if (n) 
, then the effective resistance Rn between two specified vertices of G satisfies, as n ,
 |
We only give a complete proof of this if (n) > nß, for some positive number ß. We state theorems which assert that, if (n) 
[0, ) as n , then c = 1 is a critical value of in that
if 1 then P(Rn = ) 1,
if > 1 then Rn converges to R' + R'' in distribution,
where R' and R'' are independent random variables, each of which is distributed as the electrical resistance between the root and ‘infinity’ in the family tree of a branching process whose offspring distribution is Poisson, mean , and each of whose edges has a random electrical resistance which is independent of all other edge-resistances and has distribution function F.