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Probability Theory and Related Fields
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.
The final publication is available at Springer via http://dx.doi.org/[10.1007/s00440-010-0315-6.
Montanari, A., Mossel, E., & Sly, A. (2012). The Weak Limit of Ising Models on Locally Tree-Like Graphs. Probability Theory and Related Fields, 152 (1), 31-51. http://dx.doi.org/10.1007/s00440-010-0315-6
Date Posted: 27 November 2017
This document has been peer reviewed.