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

2009/12/03 by Andrea Montanari, Montanari, Andrea, Elchanan Mossel +3 · 1 citation
Mathematics · Physics and Astronomy · #FOS: Mathematics #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Mathematical Physics (math-ph) #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.0912.0719

openalex publication_date 2009/12/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the Ising model with inverse temperature beta 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 weak 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 condition on positive magnetization (sum of spins) to the + measure on the tree.

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