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].