## The Annals of Probability

### Ferromagnetic Ising measures on large locally tree-like graphs

#### Abstract

We consider the ferromagnetic Ising model on a sequence of graphs $\mathsf{G}_{n}$ converging locally weakly to a rooted random tree. Generalizing [Probab. Theory Related Fields 152 (2012) 31–51], under an appropriate “continuity” property, we show that the Ising measures on these graphs converge locally weakly to a measure, which is obtained by first picking a random tree, and then the symmetric mixture of Ising measures with $+$ and $-$ boundary conditions on that tree. Under the extra assumptions that $\mathsf{G}_{n}$ are edge-expanders, we show that the local weak limit of the Ising measures conditioned on positive magnetization is the Ising measure with $+$ boundary condition on the limiting tree. The “continuity” property holds except possibly for countable many choices of $\beta$, which for limiting trees of minimum degree at least three, are all within certain explicitly specified compact interval. We further show the edge-expander property for (most of) the configuration model graphs corresponding to limiting (multi-type) Galton–Watson trees.

#### Article information

Source
Ann. Probab., Volume 45, Number 2 (2017), 780-823.

Dates
Revised: October 2015
First available in Project Euclid: 31 March 2017

Permanent link to this document
https://projecteuclid.org/euclid.aop/1490947308

Digital Object Identifier
doi:10.1214/15-AOP1075

Mathematical Reviews number (MathSciNet)
MR3630287

Zentralblatt MATH identifier
1372.05032

#### Citation

Basak, Anirban; Dembo, Amir. Ferromagnetic Ising measures on large locally tree-like graphs. Ann. Probab. 45 (2017), no. 2, 780--823. doi:10.1214/15-AOP1075. https://projecteuclid.org/euclid.aop/1490947308

#### References

• [1] Aizenman, M. (1980). Translation invariance and instability of phase coexistence in the two-dimensional Ising system. Comm. Math. Phys. 73 83–94.
• [2] Aizenman, M. and Wehr, J. (1990). Rounding effects of quenched randomness on first-order phase transitions. Comm. Math. Phys. 130 489–528.
• [3] Aldous, D. and Lyons, R. (2007). Processes on unimodular random networks. Electron. J. Probab. 12 1454–1508.
• [4] Aldous, D. and Steele, J. M. (2004). The objective method: Probabilistic combinatorial optimization and local weak convergence. In Probability on Discrete Structures. Encyclopaedia Math. Sci. 110 1–72. Springer, Berlin.
• [5] Athreya, K. B. and Ney, P. E. (2004). Branching Process. Dover Publications, Mineola, NY.
• [6] Benjamini, I. and Schramm, O. (2001). Recurrence of distributional limits of finite planar graphs. Electron. J. Probab. 6 13 pp. (electronic).
• [7] Berger, N., Kenyon, C., Mossel, E. and Peres, Y. (2005). Glauber dynamics on trees and hyperbolic graphs. Probab. Theory Related Fields 131 311–340.
• [8] Bodineau, T. (2006). Translation invariant Gibbs states for the Ising model. Probab. Theory Related Fields 135 153–168.
• [9] Dembo, A. and Montanari, A. (2010). Ising models on locally tree-like graphs. Ann. Appl. Probab. 20 565–592.
• [10] Dembo, A. and Montanari, A. (2010). Gibbs measures and phase transitions on sparse random graphs. Braz. J. Probab. Stat. 24 137–211.
• [11] Dembo, A., Montanari, A. and Sun, N. (2013). Factor models on locally tree-like graphs. Ann. Probab. 41 4162–4213.
• [12] De Sanctis, L. and Guerra, F. (2008). Mean field dilute ferromagnet: High temperature and zero temperature behavior. J. Stat. Phys. 132 759–785.
• [13] Dobrushin, R. L. and Shlosman, S. B. (1985). The problem of translation invariance of Gibbs states at low temperatures. In Mathematical Physics Reviews, Vol. 5. Soviet Sci. Rev. Sect. C Math. Phys. Rev. 5 53–195. Harwood Academic Publ., Chur.
• [14] Dommers, S., Giardinà, C. and van der Hofstad, R. (2010). Ising models on power-law random graphs. J. Stat. Phys. 141 638–660.
• [15] Ellis, R. S. and Newman, C. M. (1978). The statistics of Curie–Weiss models. J. Stat. Phys. 19 149–161.
• [16] Georgii, H.-O. (1988). Gibbs Measures and Phase Transitions. De Gruyter Studies in Mathematics 9. de Gruyter, Berlin.
• [17] Georgii, H.-O. and Higuchi, Y. (2000). Percolation and number of phases in the two-dimensional Ising model. J. Math. Phys. 41 1153–1169.
• [18] Greschenfield, A. and Montanari, A. (2007). Reconstruction for models on random graphs. In 48th FOCS Symposium, Providence, RI.
• [19] Griffiths, R. B., Hurst, C. A. and Sherman, S. (1970). Concavity of magnetization of an Ising ferromagnet in a positive external field. J. Math. Phys. 11 790–795.
• [20] Grimmett, G. (2006). The Random-Cluster Model. Grundlehren der Mathematischen Wissenschaften 333. Springer, Berlin.
• [21] Krengel, U. (1985). Ergodic Theorems. De Gruyter Studies in Mathematics 6. de Gruyter, Berlin.
• [22] Külske, C. (1997). Metastates in disordered mean-field models: Random field and Hopfield models. J. Stat. Phys. 88 1257–1293.
• [23] Kurtz, T., Lyons, R., Pemantle, R. and Peres, Y. (1997). A conceptual proof of the Kesten–Stigum theorem for multi-type branching processes. In Classical and Modern Branching Processes (Minneapolis, MN, 1994). IMA Vol. Math. Appl. 84 181–185. Springer, New York.
• [24] Liggett, T. M. (2005). Interacting Particle Systems. Springer, Berlin.
• [25] Lyons, R. (1989). The Ising model and percolation on trees and tree-like graphs. Comm. Math. Phys. 125 337–353.
• [26] Lyons, R. (1990). Random walks and percolation on trees. Ann. Probab. 18 931–958.
• [27] Mézard, M. and Montanari, A. (2009). Information, Physics, and Computation. Oxford Univ. Press, Oxford.
• [28] Montanari, A., Mossel, E. and Sly, A. (2012). The weak limit of Ising models on locally tree-like graphs. Probab. Theory Related Fields 152 31–51.
• [29] Newman, C. M. and Stein, D. L. (1996). Spatial inhomogeneity and thermodynamic chaos. Phys. Rev. Lett. 76 4821–4824.
• [30] Newman, M. E. J. (2003). The structure and function of complex networks. SIAM Rev. 45 167–256 (electronic).
• [31] Niss, M. (2005). History of the Lenz–Ising model 1920–1950: From ferromagnetic to cooperative phenomena. Arch. Hist. Exact Sci. 59 267–318.
• [32] Pemantle, R. and Peres, Y. (2010). The critical Ising model on trees, concave recursions and nonlinear capacity. Ann. Probab. 38 184–206.
• [33] Stroock, D. W. (2011). Probability Theory: An Analytic View, 2nd ed. Cambridge Univ. Press, Cambridge.