Multifractal spectrum of branching random walks on free groups

Consider a transient symmetric branching random walk (BRW) on a free group F indexed by a Galton-Watson tree T without leaves. The limit set is defined as the random subset of F (the boundary of F) consisting of all ends in F to which particle trajectories converge. Hueter--Lalley (2000) determined the Hausdorff dimension of the limit set, and found that ₇ 12 ₇ F. In this paper, we conduct a multifractal analysis for the limit set. We compute almost surely and simultaneously, the Hausdorff dimensions of the sets () consisting of all ends in F to which particle trajectories, with a rate of escape, converge. Moreover, for isotropic BRWs, we obtain the dimensions of the sets (, ) which consist of all ends in F to which particle trajectories, with the average rates of escape having limit points, , converge. Finally, analogous to results of Attia--Barral (2014), we obtain the Hausdorff dimensions of the level sets E (, ) of infinite branches in T along which the averages of the BRW have, as the set of accumulation points.