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 → ∞ | X n | 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.