Non-robust phase transitions in the generalized clock model on trees (Q1706298)

The authors consider spin models on locally finite trees, where the spin variables take values \(0,1,2,\dots, q-1\). The interaction between the spin variables are given by a transfer matrix which is circulant. A robust phase transition (RPT) occurs when for an arbitrary weakening of the interaction along the spins with symmetry-breaking boundary conditions suffices to induce symmetry breaking in the bulk. The authors show that for a wide range of circulant transfer matrices the sharp threshold for the existence of RPT holds for a class of finite-state spin models on general trees. For \(q=4\) and \(5\) it is shown that for generalized clock models, RPT and regular phase transition are not equivalent in general. Moreover, for two examples on a binary tree, a transition point is determined between the scenario where RPT occur at the same parameter value in terms of the third eigenvalue of the corresponding transfer matrix.