Submean variance bound for effective resistance of random electric networks

الأصل · EN

We study a model of random electric networks with Bernoulli resistances. In the case of the lattice Z², we show that the point-to-point effective resistance between 0 and a vertex v has a variance of order at most (log |v|)(2/3) whereas its expected value is of order log |v|, when v goes to infinity. When the dimension of Zᵈ is different than 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 Poincare inequality due to Falik and Samorodnitsky.