The Contact Process on Trees

Abstract

The contact process on an infinite homogeneous tree is shown to exhibit at least two phase transitions as the infection parameter $\lambda$ is varied. For small values of $\lambda$ a single infection eventually dies out. For larger $\lambda$ the infection lives forever with positive probability but eventually leaves any finite set. (The survival probability is a continuous function of $\lambda$, and the proof of this is much easier than it is for the contact process on $d$-dimensional integer lattices.) For still larger $\lambda$ the infection converges in distribution to a nontrivial invariant measure. For any $n$-ary tree, with $n$ large, the first of these transitions occurs when $\lambda \approx 1/n$ and the second occurs when $1/2\sqrt{n} < \lambda < e/\sqrt{n}$. 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 $\infty$, 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 $\lambda = 0$. The nature of the second phase transition is not yet clear and several problems are mentioned in this regard.