Limit Set Of A Weakly Supercritical Contact Process On A Homogeneous Tree

. A conjecture of Liggett [9] concerning the regime of weak survival for the contact process on a homogeneous tree is proved. The conjecture is shown to imply that the Hausdorff dimension of the limit set of such a contact process is no larger than half the Hausdorff dimension of the space of ends of the tree. The conjecture is also shown to imply that at the boundary between weak survival and strong survival, the contact process survives only weakly, a theorem previously proved by Zhang [14]. Finally, a stronger form of a theorem of Hawkes and Lyons concerning the Hausdorff dimension of a Galton-Watson tree is proved. 1. Introduction This paper concerns the growth of an isotropic contact process on an infinite homogeneous tree. The process was introduced by Pemantle [12], and discussed at some length by Liggett in the 1996 Wald Memorial Lectures [9] (see also [10]). An isotropic contact process on the homogeneous tree T = T d of degree d+1 is a continuous time Markov process A t on t..