Random Electrical Networks on Complete Graphs

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 P ( R = ∞ ) = 1 − γ ( n ) n , P ( R ⩽ x ) = γ ( n ) n F ( x ) for 0 ⩽ x < ∞ , where 0 ⩽ γ(n) ⩽ n and F is a fixed distribution function concentrated on [0, ∞). It turns out that if γ(n) → ∞, then the effective resistance Rn between two specified vertices of G satisfies, as n → ∞, y ( n ) R n → 2 { ∫ [ 0 , ∞ ] x − 1 d F ( x ) } − 1 in probability . We only give a complete proof of this if γ(n) > nβ, for some positive number β. We state theorems which assert that, if γ(n) → γ ∈ [0, ∞) as n → ∞, then γc = 1 is a critical value of γ in that if γ ⩽ 1 then P(Rn = ∞) → 1, if γ > 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 γ, and each of whose edges has a random electrical resistance which is independent of all other edge-resistances and has distribution function F.