Abstract
We obtain bounds to show that the pressure of a two-body, mean-field spin glass is a Lipschitz function of the underlying distribution of the random coupling constants, with respect to a particular semi-norm. This allows us to re-derive a result of Carmona and Hu, on the universality of the SK model, by a different proof, and to generalize this result to the Viana–Bray model. We also prove another bound, suitable when the coupling constants are not independent, which is what is necessary if one wants to consider “canonical” instead of “grand canonical” versions of the SK and Viana–Bray models. Finally, we review Viana–Bray type models, using the language of Lévy processes, which is natural in this context.
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. http://front.math.ucdavis.edu/0306.0386 (2003)
Carmona, P., Hu, Y.: Universality in Sherrington-Kirkpatrick’s spin glass model. Ann. I. H. Poincaré : PR 42(2), 215–222. http://front.math.ucdavis.edu/0403.5359 (2004)
Crawford, N.: Thermodynamics and Universality for mean field quantum spin glasses. Commun. Math. Phys. 274(3), 821–839. http://front.math.ucdavis.edu/0610.4731 (2006)
De Sanctis, L.: Random multi-overlap structures and cavity fields in diluted spin glasses. J. Stat. Phys. 117, 785–799. http://front.math.ucdavis.edu/cond-mat/0403506 (2004)
Franz, S., Leone, M.: Replica bounds for optimization problems and diluted spin systems. J. Stat. Phys. 111(3–4), 535–564. http://front.math.ucdavis.edu/0208.0280 (2003)
Guerra, F., Toninelli, F.-L.: The high temperature region of the Viana–Bray diluted spin glass model. J. Stat. Phys. 115(1–2), 531–555. http://front.math.ucdavis.edu/0302.0401 (2004)
Pastur L. and Shcherbina M. (1991). Absence of self-averaging of the order parameter in the Sherrington–Kirkpatrick model. J. Stat. Phys. 62(1–2): 1–19
Sato K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge studies in advanced mathematics, vol. 68. Cambridge University Press, UK
Viana L. and Bray A.J. (1985). Phase diagrams for dilute spin-glasses. J. Phys. C: Solid State Phys. 18: 3037–3051
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Starr, S., Vermesi, B. Some Observations for Mean-Field Spin Glass Models. Lett Math Phys 83, 281–303 (2008). https://doi.org/10.1007/s11005-008-0224-0
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11005-008-0224-0