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 Λ 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 H Λ ≤ 1/2 H ∂ F. In this paper, we conduct a multifractal analysis for the limit set Λ. We compute almost surely and simultaneously, the Hausdorff dimensions of the sets Λ(α) ⊂ Λ consisting of all ends in ∂ F to which particle trajectories, with a rate of escape α, converge. Moreover, for isotropic BRWs, we obtain the dimensions of the sets Λ(α,β) ⊂ Λ which consist of all ends in ∂ F to which particle trajectories, with the average rates of escape having limit points [α,β], converge. Finally, analogous to results of Attia--Barral (2014), we obtain the Hausdorff dimensions of the level sets E(α,β) of infinite branches in ∂ T along which the averages of the BRW have [α,β] as the set of accumulation points.