  Pemantle and Steif provided a sharp threshold for the existence of a RPT
(robust phase transition) for the continuous rotator model and the Potts model
in terms of the branching number and the second eigenvalue of the transfer
operator, where a robust phase transition is said to occur if an arbitrarily
weak coupling with symmetry-breaking boundary conditions suffices to induce
symmetry breaking in the bulk. They further showed that for the Potts model RPT
occurs at a different threshold than PT (phase transition in the sense of
multiple Gibbs measures), and conjectured that RPT and PT should occur at the
same threshold in the continuous rotator model. We consider the class of 4- and
5-state rotation-invariant spin models with reflection symmetry on general
trees which contains the Potts model and the clock model with
scalarproduct-interaction as limiting cases. The clock model can be viewed as a
particular discretization which is obtained from the classical rotator model on
the continuous one-dimensional sphere. We analyze the transition between PT=RPT
and PT is unequal to RPT, in terms of the eigenvalues of the transfer matrix of
the model at the critical threshold value for the existence of RPT. The
transition between the two regimes depends sensitively on the third largest
eigenvalue.
