Random walks on Galton-Watson trees with random conductances

We consider the random conductance model where the underlying graph is an infinite supercritical Galton-Watson tree, and the conductances are independent but their distribution may depend on the degree of the incident vertices. We prove that if the mean conductance is finite, there is a deterministic, strictly positive speed v such that lim _n→∞| ^Xn|n=v a.s. (here, |·| stands for the distance from the root). We give a formula for v in terms of the laws of certain effective conductances and show that if the conductances share the same expected value, the speed is not larger than the speed of a simple random walk on Galton-Watson trees. The proof relies on finding a reversible measure for the environment observed by the particle.