Abstract
We study the mean field dilute model of a ferromagnet. We find and prove an expression for the free energy density at high temperature, and at temperature zero. We find the critical line of the model, separating the phase with zero magnetization from the phase with symmetry breaking. We also compute exactly the entropy at temperature zero, which is strictly positive. The physical behavior at temperature zero is very interesting and related to infinite dimensional percolation, and suggests possible behaviors at generic low temperatures. Lastly, we provide a complete solution for a (partially) annealed model. Our results hold both for the Poisson and the Bernoulli versions of the model.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aizenman, M., Sims, R., Starr, S.: Extended variational principle for the Sherrington-Kirkpatrick spin-glass model. Phys. Rev. B 68, 214403 (2003)
Barra, A.: The mean field Ising model through interpolation techniques. J. Stat. Phys. (to appear). arXiv:0712.1344
Barra, A., De Sanctis, L.: Stability properties and probability distributions of multi-overlaps in diluted spin glasses. J. Stat. Mech. P08025 (2007)
Bovier, A., Gayrard, V.: The thermodynamics of the curie-weiss model with random couplings. J. Stat. Phys. 72(3–4), 643–664 (1993)
De Sanctis, L.: Random multi-overlap structures and cavity fields in diluted spin glasses. J. Stat. Phys. 117, 785–799 (2004)
De Sanctis, L.: General structures for spherical and other mean-field spin models. J. Stat. Phys. 126, 817–835 (2006)
Dorogovtsev, S.N., Goltsev, A.V., Mendes, J.F.F.: Critical phenomena in complex networks. arXiv:cond-mat/0705.0010 (2007)
Franz, S., Leone, M.: Replica bounds for optimization problems and diluted spin systems. J. Stat. Phys. 111, 535–564 (2003)
Gerschenfeld, A., Montanari, A.: Reconstruction for models on random graphs. In: Proc. Foun. of Comp. Sci. (2007)
Guerra, F.: About the overlap distribution in mean field spin glass models. Int. J. Mod. Phys. B 10, 1675–1684 (1996)
Guerra, F.: Sum rules for the free energy in the mean field spin glass model. Fields Inst. Commun. 30, 161 (2001)
Guerra, F.: Broken replica symmetry bounds in the mean field spin glass model. Commun. Math. Phys. 233, 1–12 (2003)
Guerra, F.: An introduction to mean field spin glass theory: methods and results. In: Bovier, A. et al. (eds.) Mathematical Statistical Physics, pp. 243–271. Elsevier, Amsterdam (2006)
Guerra, F., Toninelli, F.L.: The thermodynamic limit in mean field spin glass models. Commun. Math. Phys. 230, 71–79 (2002)
Guerra, F., Toninelli, F.L.: The high temperature region of the Viana-Bray diluted spin glass model. J. Stat. Phys. 115, 531–555 (2004)
Hase, M.O., de Almeida, J.R.L., Salinas, S.R.: Relica-symmetric solutions of a dilute Ising ferromagnet in a random field. Eur. Phys. J. B 47, 245–249 (2005)
Janson, S., Luczak, T., Rucinski, A.: Random Graphs. Wiley, New York (2000)
Ostilli, M.: Ising spin glass models versus Ising models: an effective mapping at high temperature: I. General result. J. Stat. Mech. P10004 (2006)
Kanter, I., Sompolinsky, H.: Mean-field theory of spin-glass with finite coordination number. Phys. Rev. Lett. 58(2), 164–167 (1987)
Starr, S.L., Vermesi, B.: Some observations for mean-field spin glass models. arXiv:0707.0031 (2007)
Talagrand, M.: Spin Glasses: A Challenge for Mathematicians. Cavity and Mean Field Models. Springer, Berlin (2003)
Talagrand, M.: The Parisi formula. Ann. Math. 163, 221–263 (2006)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
De Sanctis, L., Guerra, F. Mean Field Dilute Ferromagnet: High Temperature and Zero Temperature Behavior. J Stat Phys 132, 759–785 (2008). https://doi.org/10.1007/s10955-008-9575-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-008-9575-2