Metric and ultrametric spaces of resistances

Given an electrical circuit each edge e of which is an isotropic conductor with a monomial conductivity function y∗ e = y r e/μ s e. In this formula, ye is the potential difference and y ∗ e current in e, while μe is the resistance of e; furthermore, r and s are two strictly positive real parameters common for all edges. In particular, the case r = s = 1 corresponds to the standard Ohm law. In 1987, Gvishiani and Gurvich [Russ. Math. Surveys, 42:6(258) (1987) 235–236] proved that, for every two nodes a, b of the circuit, the effective resistance μa,b is well-defined and for every three nodes a, b, c the inequality μ s/r a,b ≤ μ s/r a,c + μ s/r c,b holds. It obviously implies the standard triangle inequality μa,b ≤ μa,c + μc,b whenever s ≥ r. For the case s = r = 1, these results were rediscovered in 1990s. Now, in 23 years, I venture to reproduce the proof of the original result for the following reasons: • It is more general than just the case r = s = 1 and one can get several interesting metric and ultrametric spaces playing with parameters r and s. In particular, (i) the effective Ohm resistance, (ii) the length of a shortest path, (iii) the inverse width of a bottleneck path, and (iv) the inverse capacity (maximum flow per unit time) between any pair of terminals a and b provide four examples of the resistance distances μa,b that can be obtained from the above model by the following limit transitions: (i) r(t) = s(t) ≡ 1, (ii) r(t) = s(t)→∞, (iii) r(t) ≡ 1, s(t)→∞, and (iv) r(t)→ 0, s(t) ≡ 1, as t → ∞. In all four cases the limits μa,b = limt→∞ μa,b(t) exist for all pairs a, b and the metric inequality μa,b ≤ μa,c + μc,b holds for all triplets a, b, c, since s(t) ≥ r(t) for any sufficiently large t. Moreover, the stronger ultrametric inequality μa,b ≤ max(μa,c, μc,b) holds for all triplets a, b, c in examples (iii) and (iv), since in these two cases s(t)/r(t)→∞, as t→∞. • Communications of the Moscow Math. Soc. in Russ. Math. Surveys were (and still are) strictly limited to two pages; the present paper is much more detailed. Although translation in English of the Russ. Math. Surveys is available, it is not free in the web and not that easy to find out. • The last but not least: priority.