Metric and ultrametric spaces of resistances

Abstract

Consider an electrical circuit, each edge e of which is an isotropic conductor with a monomial conductivity function y e ∗ = y e r / μ e s . In this formula, y e is the potential difference and y e ∗ current in e , while μ e 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’s law. In 1987, Gvishiani and Gurvich [A.D. Gvishiani, V.A. Gurvich, Metric and ultrametric spaces of resistances, in: Communications of the Moscow Mathematical Society, Russian Math. Surveys 42 (6 (258)) (1987) 235–236] proved that, for every two nodes a , b of the circuit, the effective resistance μ a , b is well-defined and for every three nodes a , b , c the inequality μ a , b s / r ≤ μ a , c s / r + μ c , b s / r holds. It obviously implies the standard triangle inequality μ a , b ≤ μ a , c + μ c , b whenever s ≥ r . For the case s = r = 1 , these results were rediscovered in the 1990s. Now, after 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 μ a , 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 μ a , b = lim t → ∞ μ a , b ( t ) exist for all pairs a , b and the metric inequality μ a , b ≤ μ a , c + μ c , b holds for all triplets a , b , c , since s ( t ) ≥ r ( t ) for any sufficiently large t . Moreover, the stronger ultrametric inequality μ a , b ≤ max ( μ a , c , μ c , 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 a translation in English of the Russ. Math. Surveys is available, it is not free in the web and not that easy to find. • The last but not least: priority.