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Pemantle and Steif provided a sharp threshold for the existence of a RPT(robust phase transition) for the continuous rotator model and the Potts modelin terms of the branching number and the second eigenvalue of the transferoperator, where a robust phase transition is said to occur if an arbitrarilyweak coupling with symmetry-breaking boundary conditions suffices to inducesymmetry breaking in the bulk. They further showed that for the Potts model RPToccurs at a different threshold than PT (phase transition in the sense ofmultiple Gibbs measures), and conjectured that RPT and PT should occur at thesame threshold in the continuous rotator model. We consider the class of 4- and5-state rotation-invariant spin models with reflection symmetry on generaltrees which contains the Potts model and the clock model withscalarproduct-interaction as limiting cases. The clock model can be viewed as aparticular discretization which is obtained from the classical rotator model onthe continuous one-dimensional sphere. We analyze the transition between PT=RPTand PT is unequal to RPT, in terms of the eigenvalues of the transfer matrix ofthe model at the critical threshold value for the existence of RPT. Thetransition between the two regimes depends sensitively on the third largesteigenvalue.

Non-robust phase transitions in the generalized clock model on trees