The Annals of Applied Probability
- Ann. Appl. Probab.
- Volume 20, Number 2 (2010), 565-592.
Ising models on locally tree-like graphs
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.
Ann. Appl. Probab., Volume 20, Number 2 (2010), 565-592.
First available in Project Euclid: 9 March 2010
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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
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