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

## TL;DR

This paper investigates the conditions under which robust phase transitions occur in generalized clock models on trees, revealing a sensitive dependence on eigenvalues of the transfer matrix and distinguishing between PT and RPT thresholds.

## Contribution

It extends the analysis of phase transition thresholds to 4- and 5-state models, highlighting the role of eigenvalues in the transition between PT=RPT and PT≠RPT regimes.

## Key findings

- Transition between PT=RPT and PT≠RPT depends on the third largest eigenvalue.
- RPT and PT thresholds differ in certain models, contrary to previous conjectures.
- Eigenvalues of the transfer matrix critically determine phase transition behavior.

## Abstract

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.