The Weak Survival/strong Survival Phase Transition for the Contact Process on a Homogeneous Tree

The contact process on a homogeneous tree Td of degree d + 1 ≥ 3 is known [10, 7, 13] to have three distinct phases: an extinction phase, a weak survival phase, and a strong survival phase. The existence of two qualitatively different survival phases is the most striking feature of the process, as the contact process on the integer lattice Zd, in any dimension, exhibits only one survival phase (strong survival). Thus, the contact process on a homogeneous tree exhibits a phase transition, from weak to strong survival, of a different character than the phase transition for the contact process on the integer lattices. The purpose of this paper is to speculate on the nature of this phase transition, and to show how certain conjectured behavior of the expected total infection time in the weak survival phase would delimit the critical exponent of the “Malthusian parameter” βd defined by (1) below. In the weak survival phase, the contact process, when started from a single infected site (by convention, the root vertex r of the tree), survives forever with positive probability, but with probability one eventually vacates every finite subset of the tree. For any vertex x other than the root, the probability of eventual infection is less than one. This probability ux = un depends only on the distance n = |x| from r to x, and decays exponentially in n; the decay rate is