Abstract
We consider the Ising model with inverse temperature β and without external field on sequences of graphs G n which converge locally to the k-regular tree. We show that for such graphs the Ising measure locally weakly converges to the symmetric mixture of the Ising model with + boundary conditions and the − boundary conditions on the k-regular tree with inverse temperature β. In the case where the graphs G n are expanders we derive a more detailed understanding by showing convergence of the Ising measure conditional on positive magnetization (sum of spins) to the + measure on the tree.
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Montanari, A., Mossel, E. & Sly, A. The weak limit of Ising models on locally tree-like graphs. Probab. Theory Relat. Fields 152, 31–51 (2012). https://doi.org/10.1007/s00440-010-0315-6
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DOI: https://doi.org/10.1007/s00440-010-0315-6
Mathematics Subject Classification (2000)
- 60K35
- 82B20
- 82B26