THE RANGE OF TREE-INDEXED RANDOM WALK. (10th September 2014) - Le Gall, Jean-François; Lin, Shen
Abstract : We provide asymptotics for the range R_n of a random walk on the d -dimensional lattice indexed by a random tree with n vertices. Using Kingman's subadditive ergodic theorem, we prove under general assumptions that n^-1R_n converges to a constant, and we give conditions ensuring that the limiting constant is strictly positive. On the other hand, in dimension 4, and in the case of a symmetric random walk with exponential moments, we prove that R_n grows like n/\!n . We apply our results to asymptotics for the range of a branching random walk when the initial size of the population tends to infinity.