Robust phase transitions for Heisenberg and other models on general trees

We study several statistical mechanical models on a general tree. Particula r attention is devoted to the classical Heisenberg models, where the state space is the d-dimensional unit sphere and the interactions are proportiona l to the cosines of the angles between neighboring spins. The phenomenon of interest here is the classification of phase transition (nonuniqueness of the Gibbs state) according to whether it is robust. in many cases, includin g all of the Heisenberg and Potts models, occurrence of robust phase transi tion is determined by the geometry (branching number) of the tree in a way that parallels the situation with independent percolation and usual phase t ransition for the Ising model. The critical values for robust phase transit ion for the Heisenberg and Potts models are also calculated exactly. In som e cases, such as the q greater than or equal to 3 Potts model, robust phase transition and usual phase transition do not coincide, while in other case s, such as the Heisenberg models, we conjecture that robust phase transitio n and usual phase transition are equivalent. In addition, we show that symm etry breaking is equivalent to the existence of a phase transition, a fact believed but not known for the rotor model on Z(2).