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April 1999 Robust Phase Transitions for Heisenberg and other Models on General Trees
Robin Pemantle, Jeffrey E. Steif
Ann. Probab. 27(2): 876-912 (April 1999). DOI: 10.1214/aop/1022677389


We study several statistical mechanical models on a general tree. Particular attention is devoted to the classical Heisenberg models, where the state space is the $d$-dimensional unit sphere and the interactions are proportional to the cosines of the angles between neighboring spins. The phenomenon of interest here is the classification of phase transition (non-uniqueness of the Gibbs state) according to whether it is robust. In many cases, including all of the Heisenberg and Potts models, occurrence of robust phase transition is determined by the geometry (branching number) of the tree in a way that parallels the situation with independent percolation and usual phase transition for the Ising model. The critical values for robust phase transition for the Heisenberg and Potts models are also calculated exactly. In some cases, such as the $q \geq 3$ Potts model, robust phase transition and usual phase transition do not coincide, while in other cases, such as the Heisenberg models, we conjecture that robust phase transition and usual phase transition are equivalent. In addition, we show that symmetry breaking is equivalent to the existence of a phase transition, a fact believed but not known for the rotor model on $\mathbb{Z}^2$ .


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Robin Pemantle. Jeffrey E. Steif. "Robust Phase Transitions for Heisenberg and other Models on General Trees." Ann. Probab. 27 (2) 876 - 912, April 1999.


Published: April 1999
First available in Project Euclid: 29 May 2002

zbMATH: 0981.60096
MathSciNet: MR1698979
Digital Object Identifier: 10.1214/aop/1022677389

Primary: 60K35 , 82B05 , 82B26

Keywords: Heisenberg models , Phase transitions , symmetry breaking

Rights: Copyright © 1999 Institute of Mathematical Statistics


Vol.27 • No. 2 • April 1999
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