Continuous spin models on annealed generalized random graphs

Dommers, Sander; Kuelske, Christof; Philipp, Schriever,

doi:10.48550/arxiv.1610.08242

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“…In these models the random graphs are often assumed (or proven in the largegraph limit) to be locally tree-like. To understand the behavior of these systems it is important to understand the situation on the tree first [4][5][6][7].…”

Section: Introductionmentioning

confidence: 99%

Non-robust Phase Transitions in the Generalized Clock Model on Trees

Külske¹,

Philipp²

2017

J Stat Phys

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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 symmetrybreaking 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 with state space S 1 . We analyze the transition between PT=RPT and PT =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.Mathematics Subject Classifications (2010). 82B26 (primary); 60K35 (secondary)

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Section: Introductionmentioning

confidence: 99%