RANDOM ELECTRICAL NETWORKS ON COMPLETE GRAPHS

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 Kn+2 when each edge is allocated a random resistance. These resistances are assumed independent with distribution P(R = ∞) = 1 − nγ(n), P(R ≤ x) = nγ(n)F (x) for 0 ≤ x < ∞, where F is a fixed distribution function and γ(n) → γ ≥ 0 as n → ∞. The asymptotic effective resistance between two chosen vertices is identified in the two cases γ ≤ 1 and γ > 1, and the case γ = ∞ is considered. The analysis proceeds via detailed estimates based on the theory of branching processes.