We work on a Galton-Watson tree with random weights, in the so-called "subdiffusive" regime. We study the rate of decay of the conductance between the root and the n-th level of the tree, as n goes to infinity, by a mostly analytic method. It turns out the order of magnitude of the expectation of this conductance can be less than 1/n (in contrast with the results of Addario-Berry-Broutin-Lugosi and Chen-Hu-Lin), depending on the value of the second zero of the characteristic function associated to the model.

We also prove the almost sure (and in L p for some p > 1) convergence of this conductance divided by its expectation towards the limit of the additive martingale.