The existence of an intermediate phase for the contact process on trees

Let T d be a homogeneous tree in which every vertex has d neighbors. A new proof is given that the contact process on T d exhibits two phase transitions when d ≥ 3, a behavior which distinguishes it from the contact process on Z n . This is the first proof which does not involve calculation of bounds on critical values, and it is much shorter than the previous proof for the binary tree, T 3 . The method is extended to prove the existence of an intermediate phase for a more general class of trees with exponential growth and certain symmetry properties, for which no such result was previously known.