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 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 an n-ary tree, with n large, the first of these transitions occurs when lambda~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. 
Robin Pemantle