  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.
