Random Electrical Networks on Complete Graphs II: Proofs

Abstract

This paper contains the proofs of Theorems 2 and 3 of the article entitled Random Electrical Networks on Complete Graphs, written by the same authors and published in the Journal of the London Mathematical Society, vol. 30 (1984), pp. 171-192. The current paper was written in 1983 but was not published in a journal, although its existence was announced in the LMS paper. This TeX version was created on 9 July 2001. It incorporates minor improvements to formatting and punctuation, but no change has been made to the mathematics. We study the effective electrical resistance of the complete graph $K_{n+2}$ when each edge is allocated a random resistance. These resistances are assumed independent with distribution $P(R=\infty)=1-n^{-1}\gamma(n)$, $P(R\le x) = n^{-1}\gamma(n)F(x)$ for $0\le x < \infty$, where $F$ is a fixed distribution function and $\gamma(n)\to\gamma\ge 0$ as $n\to\infty$. The asymptotic effective resistance between two chosen vertices is identified in the two cases $\gamma\le 1$ and $\gamma>1$, and the case $\gamma=\infty$ is considered. The analysis proceeds via detailed estimates based on the theory of branching processes.