Effective resistance on random electrical networks

Take a big graph and make a random electrical network of it by assigning independent resistances on its edges. Now, ask for the behaviour of the effective resistance between two vertices (two “poles”) far apart. We assume in general that resistances are bounded away from 0 and ∞. In this paper, we study three cases of effective resistance in such random electrical networks: from one side to another in a box of Z, between two points in Z, and between two points on a cylinder graph G×Z. For all these cases, we obtain the right order of the fluctuations when the poles move apart from each other, and give corresponding subgaussian concentration inequalities. For the cylinder graphs, we prove two additional results: a central limit theorem and a result of uniform stability with respect to noise.