The Contact Process on Trees

The contact process on an infinite homogeneous tree is shown to exhibit at least two phase transitions as the infection parameter λ is varied. For small values of λ a single infection eventually dies out. For larger λ the infection lives forever with positive probability but eventually leaves any finite set. (The survival probability is a continuous function of λ, and the proof of this is much easier than it is for the contact process on d-dimensional integer lattices.) For still larger λ the infection converges in distribution to a nontrivial invariant measure. For an n-ary tree, with n large, the first of these transitions occurs when λ ≈ 1/n and the second occurs when 1/2 √ n < λ < e/ √ n. Nonhomogeneous trees whose vertices have degrees varying between 1 and n behave essentially as homogeneous n-ary trees, provided that vertices of degree n are not too rare. In particular, letting n go to ∞, Galton-Watson trees whose vertices have degree n with probability that does not decrease exponentially with n may have both phase transitions occur together at λ = 0. The nature of the second phase transition is not yet clear and several problems are mentioned in this regard.