The Annals of Probability

On the overlap in the multiple spherical SK models

Dmitry Panchenko and Michel Talagrand

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In order to study certain questions concerning the distribution of the overlap in Sherrington–Kirkpatrick type models, such as the chaos and ultrametricity problems, it seems natural to study the free energy of multiple systems with constrained overlaps. One can write analogues of Guerra’s replica symmetry breaking bound for such systems but it is not at all obvious how to choose informative functional order parameters in these bounds. We were able to make some progress for spherical pure p-spin SK models where many computations can be made explicitly. For pure 2-spin model we prove ultrametricity and chaos in an external field. For the pure p-spin model for even p>4 without an external field we describe two possible values of the overlap of two systems at different temperatures. We also prove a somewhat unexpected result which shows that in the 2-spin model the support of the joint overlap distribution is not always witnessed at the level of the free energy and, for example, ultrametricity holds only in a weak sense.

Article information

Ann. Probab., Volume 35, Number 6 (2007), 2321-2355.

First available in Project Euclid: 8 October 2007

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

Spin glasses free energy chaos ultrametricity


Panchenko, Dmitry; Talagrand, Michel. On the overlap in the multiple spherical SK models. Ann. Probab. 35 (2007), no. 6, 2321--2355. doi:10.1214/009117907000000015.

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  • Crisanti, A. and Sommers, H. J. (1992). The spherical $p$-spin interaction spin glass model: The statics. Z. Phys. B. Condensed Matter 83 341--354.
  • Ben Arous, G., Dembo, A. and Guionnet, A. (2001). Aging of spherical spin glasses. Probab. Theory Related Fields 120 1--67.
  • Guerra, F. (2003). Broken replica symmetry bounds in the mean field spin glass model. Comm. Math. Phys. 233 1--12.
  • Panchenko, D. (2005). Free energy in the generalized Sherrington--Kirkpatrick mean field model. Rev. Math. Phys. 17 793--857.
  • Sherrington, D. and Kirkpatrick, S. (1972). Solvable model of a spin glass. Phys. Rev. Lett. 35 1792--1796.
  • Talagrand, M. (2003). Spin Glasses: a Challenge for Mathematicians. Cavity and Mean Field Models. Springer, Berlin.
  • Talagrand, M. (2006). The Parisi formula. Ann. of Math. (2) 163 221--263.
  • Talagrand, M. (2006). Free energy of the spherical mean field model. Probab. Theory Related Fields 134 339--382.
  • Talagrand, M. (2007). Mean field models for spin glasses: Some obnoxious problems. Proceedings of the Conference ``Mathematical Physics of Spin Glasses'' Cortona, Italy, 2005. To appear.