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The weak limit of Ising models on locally tree-like graphs
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  • Published: 11 September 2010

The weak limit of Ising models on locally tree-like graphs

  • Andrea Montanari1,2,
  • Elchanan Mossel3,4 &
  • Allan Sly5 

Probability Theory and Related Fields volume 152, pages 31–51 (2012)Cite this article

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  • 26 Citations

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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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Author information

Authors and Affiliations

  1. Department of Electrical Engineering, Stanford University, Stanford, CA, 94305-9510, USA

    Andrea Montanari

  2. Department of Statistics, Stanford University, Stanford, CA, 94305-9510, USA

    Andrea Montanari

  3. Faculty of Mathematics and Computer Science, Weizmann Institute, Rehovot, Israel

    Elchanan Mossel

  4. Departments of Statistics and Computer Science, UC Berkeley, Berkeley, CA, 94720-3860, USA

    Elchanan Mossel

  5. Microsoft Research, Redmond, WA, 98052, USA

    Allan Sly

Authors
  1. Andrea Montanari
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  2. Elchanan Mossel
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  3. Allan Sly
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Correspondence to Andrea Montanari.

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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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  • Received: 03 December 2009

  • Revised: 06 June 2010

  • Published: 11 September 2010

  • Issue Date: February 2012

  • DOI: https://doi.org/10.1007/s00440-010-0315-6

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Mathematics Subject Classification (2000)

  • 60K35
  • 82B20
  • 82B26
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