A conjecture of Liggett 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. Finally, a stronger form of a theorem of Hawkes and Lyons concerning the Hausdorff dimension of a Galton–Watson tree is proved.
"Limit set of a weakly supercritical contact process on a homogeneous tree." Ann. Probab. 26 (2) 644 - 657, April 1998. https://doi.org/10.1214/aop/1022855646