Capacity of the range of tree-indexed random walk.

We provide the asymptotics of the capacity of a random walk indexed by an infinite Galton-Watson tree. In the first place we prove that the capacity grows linearly in dimensions $d\ge 7$. And secondly by introducing a new measure on infinite random planar trees, we show that the capacity grows like $n\log n$ in the critical dimension $d=6$. Using our results and the framework established in Le Gall and Lin [14] on conditioned trees, we are able to deduce an analogue of the convergence for random walk indexed by uniform rooted random planar tree with $n$ vertices.