Open Access
November 2007 On the overlap in the multiple spherical SK models
Dmitry Panchenko, Michel Talagrand
Ann. Probab. 35(6): 2321-2355 (November 2007). DOI: 10.1214/009117907000000015


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.


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Dmitry Panchenko. Michel Talagrand. "On the overlap in the multiple spherical SK models." Ann. Probab. 35 (6) 2321 - 2355, November 2007.


Published: November 2007
First available in Project Euclid: 8 October 2007

zbMATH: 1128.60086
MathSciNet: MR2353390
Digital Object Identifier: 10.1214/009117907000000015

Primary: 60K35 , 82B44

Keywords: chaos , Free energy , Spin glasses , Ultrametricity

Rights: Copyright © 2007 Institute of Mathematical Statistics

Vol.35 • No. 6 • November 2007
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