The Annals of Applied Probability

Ising models on locally tree-like graphs

Amir Dembo and Andrea Montanari

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Abstract

We consider ferromagnetic Ising models on graphs that converge locally to trees. Examples include random regular graphs with bounded degree and uniformly random graphs with bounded average degree. We prove that the “cavity” prediction for the limiting free energy per spin is correct for any positive temperature and external field. Further, local marginals can be approximated by iterating a set of mean field (cavity) equations. Both results are achieved by proving the local convergence of the Boltzmann distribution on the original graph to the Boltzmann distribution on the appropriate infinite random tree.

Article information

Source
Ann. Appl. Probab., Volume 20, Number 2 (2010), 565-592.

Dates
First available in Project Euclid: 9 March 2010

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1268143433

Digital Object Identifier
doi:10.1214/09-AAP627

Mathematical Reviews number (MathSciNet)
MR2650042

Zentralblatt MATH identifier
1191.82025

Subjects
Primary: 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)
Secondary: 82B23: Exactly solvable models; Bethe ansatz 60F10: Large deviations 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 05C80: Random graphs [See also 60B20] 05C05: Trees

Keywords
Ising model random sparse graphs cavity method Bethe measures belief propagation local weak convergence

Citation

Dembo, Amir; Montanari, Andrea. Ising models on locally tree-like graphs. Ann. Appl. Probab. 20 (2010), no. 2, 565--592. doi:10.1214/09-AAP627. https://projecteuclid.org/euclid.aoap/1268143433


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