Random Electrical Networks on Complete Graphs II: Proofs

arXivmath/0107068MaRDI QIDQ6471202

Harry Kesten, Geoffrey R. Grimmett

Publication date: 10 July 2001

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 $P(R=infty)=1-n^{-1}gamma(n)$, $P(Rlex)=n^{-1}gamma(n)F(x)$ for $0lex<infty$, where $F$ is a fixed distribution function and $gamma(n)ogammage0$ as $noinfty$. The asymptotic effective resistance between two chosen vertices is identified in the two cases $gammale1$ and $gamma>1$, and the case $gamma=infty$ is considered. The analysis proceeds via detailed estimates based on the theory of branching processes.

Mathematics Subject Classification ID

Interacting random processes; statistical mechanics type models; percolation theory (60K35) Percolation (82B43)

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