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    188宝金博页面版: On a nonhierarchical version of the Generalized Random Energy Model, II Ultrametricity

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    内容提示： Stochastic Processes and their Applications 119 (2009) 2357–2386www.elsevier.com/locate/spaOn a nonhierarchical version of the GeneralizedRandom Energy Model, II: UltrametricityErwin Bolthausen a , Nicola Kistler b,?a Universit?t Zürich, Winterthurerstrasse 190, CH-8057 Zurich, Switzerlandb Universit?t Bonn, Wegelerstr. 6, DE-53115 Bonn, GermanyReceived 16 February 2008; received in revised form 18 November 2008; accepted 10 December 2008Available online 16 December 2008AbstractWe study the Gibbs meas...

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    Stochastic Processes and their Applications 119 (2009) 2357–2386www.elsevier.com/locate/spaOn a nonhierarchical version of the GeneralizedRandom Energy Model, II: UltrametricityErwin Bolthausen a , Nicola Kistler b,∗a Universität Zürich, Winterthurerstrasse 190, CH-8057 Zurich, Switzerlandb Universität Bonn, Wegelerstr. 6, DE-53115 Bonn, GermanyReceived 16 February 2008; received in revised form 18 November 2008; accepted 10 December 2008Available online 16 December 2008AbstractWe study the Gibbs measure of the nonhierarchical versions of the Generalized Random Energy Modelsintroduced in previous work. We prove that the ultrametricity holds only provided some nondegeneracyconditions on the Hamiltonian are met.c ? 2008 Elsevier B.V. All rights reserved.Keywords: Disordered systems; Spin glasses; Ultrametricity; Extreme values1. IntroductionThe study of spin glasses, a paradigm for the statistical mechanics of disordered systems, hasattracted a lot of interest ever since their introduction in the field of condensed matter. Given thesuccess of the Ising model for an understanding of basic questions in statistical physics, probablythe most natural spin glass model is the Edwards–Anderson model which is a spin model withlattice Z d , and random nearest neighbor interactions. Mathematically, this model remains to thisday, totally intractable. The situation is much better for the Sherrington–Kirkpatrick model (SKfor short), which is of mean-field type, meaning that every spin interacts with any other on equalfooting. For the SK-model, a marvellous theory was introduced by Giorgio Parisi in the 1970s, cf.for more on this [1], which has been further developed by many authors. This is a fully developedtheorywhichhassuccessfullybeenappliedtomanyotherproblems,forinstanceincombinatorialoptimization, but there was no mathematically rigorous foundation, till quite recently.∗ Corresponding address: Abt. Wahrscheinlichkeitstheorie Institut fur Angewandte Mathematik, Wegelerstr. 6, D-53115 Bonn, Germany. Tel.: +49 228 733417; fax: +49 228 739537.E-mail addresses: eb@math.uzh.ch (E. Bolthausen), nkistler@wiener.iam.uni-bonn.de (N. Kistler).0304-4149/$ - see front matter c ? 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.spa.2008.12.002

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