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Particle systems on trees have produced the first and most tractable examples of certain qualitative phenomena. For example, the contact process on a tree has multiple phase transitions, ([19, 12, 22]) and the critical temperature for the Ising model on a tree is determined by its branching number or Hausdorff dimension ([13, 8, 20]), which makes the Ising model intimately related to independent percolation whose critical value is also determined by the branching number (see [14]). In this paper we study several models on general infinite trees, including the classical Heisenberg and Potts models. Our aim is to exhibit a distinction between two kinds of phase transitions, robust and non-robust, as well as to investigate conditions under which robust phase transitions occur.

In many cases, including the Heisenberg and Potts models, the existence of a robust phase transition is determined by the branching number. However, in some cases (in- cluding the q > 2 Potts model), the critical temperature for the existence of usual phase transition is not determined by the branching number. Thus robust phase transition be- haves in a more universal manner than non-robust phase transition, being a function of the branching number alone, as it is for usual phase transition for independent percola- tion and the Ising model. Although particle systems on trees do not always predict the qualitative behavior of the same particle system on high-dimensional lattices, it seems likely that there is a lattice analogue of non-robust phase transition, which would make an interesting topic for further res