Submean Variance Bound for Effective Resistance of Random Electric Networks

We study a model of random electric networks with Bernoulli resistances. In the case of the lattice \({\mathbb{Z}^2}\) , we show that the point-to-point effective resistance between 0 and a vertex v has a variance of order at most \({({\rm log} |v|)^{\frac{2}{3}}}\) , whereas its expected value is of order log | v|, when v goes to infinity. When d ≠ 2, expectation and variance are of the same order. Similar results are obtained in the context of p- resistance. The proofs rely on a modified Poincaré inequality due to Falik and Samorodnitsky 7].