A Note on the Guerra and Talagrand Theorems for Mean Field Spin Glasses: The Simple Case of Spherical Models

Franz, Silvio; Tria, Francesca

doi:10.1007/s10955-005-8007-9

Cited by 6 publications

(8 citation statements)

References 24 publications

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“…It is worthwhile to remark that the explicit representation (12) coincides with the RS approximation exhibited in [4]. This enables us to complete the picture, by identifying q with the Edward-Anderson order parameter of the model.…”

Section: Conclusion and Outlooksmentioning

confidence: 66%

“…Since it depends in fact by R 2 , we will also denote the pressure as A sf (β, R 2 ) (in the literature one finds λ, R = 1). The results of Crisanti, Sommers and Talagrand, along with later works on the subject ( [20][19] [12]), deal with Gaussian disorder. However we will see that this assumption can be relaxed in the sense of hypothesis H (see also [6]).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Legendre Duality of Spherical and Gaussian Spin Glasses

Genovese

Tantari

2015

Math Phys Anal Geom

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Abstract. The classical result of concentration of the Gaussian measure on the sphere in the limit of large dimension induces a natural duality between Gaussian and spherical models of spin glass. We analyse the Legendre variational structure linking the free energies of these two systems, in the spirit of the equivalence of ensembles of statistical mechanics. Our analysis, combined with the previous work [4], shows that such models are replica symmetric. Lastly, we briefly discuss an application of our result to the study of the Gaussian Hopfield model. MSC: 82B44.

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Section: Conclusion and Outlooksmentioning

confidence: 66%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Legendre Duality of Spherical and Gaussian Spin Glasses

Genovese

Tantari

2015

Math Phys Anal Geom

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“…Analytical computations, and in particular expansions around the transition line, can be performed explicitly in this model and have constituted a very useful guideline for the analysis of the quantum coloring model. For these reasons we briefly collect in this Appendix the main results on this well-known model that are enlightening with this application in mind and refer the reader to [65,66,[70][71][72][73] for more details on this model. Its Hamiltonian is…”

Section: Discussionmentioning

confidence: 99%

Effect of quantum fluctuations on the coloring of random graphs

2013

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We present a study of the coloring problem (antiferromagnetic Potts model) of random regular graphs, submitted to quantum fluctuations induced by a transverse field, using the quantum cavity method and quantum Monte-Carlo simulations. We determine the order of the quantum phase transition encountered at low temperature as a function of the transverse field and discuss the structure of the quantum spin glass phase. In particular, we conclude that the quantum adiabatic algorithm would fail to solve efficiently typical instances of these problems because of avoided level crossings within the quantum spin glass phase, caused by a competition between energetic and entropic effects.

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“…Actually, the extremum is a maximum, as can be seen by an explicit computation. Remarkably, the fact that the equilibrium spin glass free energy is the maximum of φ 1RSB with respect to m and q is the usual prescription of the replica method, and it can be rigorously proven to be correct [16]. In summary, the replica method allowed us to fully characterize the thermodynamics of the spherical p-spin model, by computing…”

Section: The Phase Diagram Of the Spherical P-spin Modelmentioning

confidence: 99%

“…• Spin glasses also provide a testing ground for a more mathematically inclined probabilistic approach: the rigorous proof of the correctness of the solution of the mean field model came out after twenty years of efforts where new ideas, e.g. new variational principles [14], were at the basis of a recent rigorous proof (see [15] and [16] for a concise explanation of the main ideas) of the correctness of the mean field approximation in the case of the infinite range Sherrington-Kirkpatrick and p-spin models that we will introduce below.…”

Section: Introductionmentioning

confidence: 99%

Mean field theory of spin glasses

Zamponi¹

2010

Preprint

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These lecture notes focus on the mean field theory of spin glasses, with particular emphasis on the presence of a very large number of metastable states in these systems. This phenomenon, and some of its physical consequences, will be discussed in details for fully-connected models and for models defined on random lattices. This will be done using the replica and cavity methods.These notes have been prepared for a course of the PhD program in Statistical Mechanics at SISSA, Trieste and at the University of Rome "Sapienza". Part of the material is reprinted from other lecture notes, and when this is done a reference is obviously provided to the original. I would like to warmly thank all the students and colleagues who read these notes, gave me their feedback and sent me their corrections, that allowed to fix many errors on the original manuscript. I also would like to thank SISSA and the University of Rome "Sapienza" for inviting me to give these lectures. Contents I. IntroductionA. Why study spin glasses? B. Physical systems C. Optimization problems D. Models and universality classes E. Frustration and quenched disorder F. What is missing in these notes II. Fully connected models A. Free energy functional 1. The fully connected Ising ferromagnet 2. Metastable states in fully connected models 3. The general definition of the free energy functional 4. The Georges-Yedidia expansion 5. Free energy functional for a generic Ising model 6. Back to the fully connected ferromagnet 7. TAP equations for the SK model 8. Spherical p-spin model 9. Summary and remarks B. Metastable states and complexity 1. The simplest example: the spherical p-spin glass model 2. The partition function 3. A method to compute the complexity 4. Replicated free energy of the spherical p-spin model 5. 1-step replica symmetry breaking 6. The phase diagram of the spherical p-spin model 7. Spontaneous replica symmetry breaking: the order parameter C. The SK model: full replica symmetry breaking 1. The overlap distribution 2. The Parisi solution of the SK model D. Susceptibilities 1. The ferromagnet 2. Spin glasses: linear susceptibilities 3. The static spin glass susceptibility 4. The dynamic spin glass susceptibility E. Exercises III. Diluted models and optimization problems A. Definitions 1. Statistical mechanics formulation of optimization problems 2. Random optimization problems: random graphs and hypergraphs 3. Connectivity-temperature phase diagram B. XORSAT: clustering and SAT/UNSAT transition 1. Bounds from the first and second moments methods 2. The leaf-removal algorithm 3. Clustering and SAT/UNSAT transitions. 4. On backbones 5. Summary C. The replica symmetric cavity method 1. Recursions on a finite tree 2. From a tree to a random graph 3. Replica symmetric cavity equations in absence of local disorder 4. An alternative derivation 5. Fluctuations of the local environment: distributions of cavity probabilities 6. The zero temperature limit 7. On a factor graph 8. Summary D. 1-step replica symmetry breaking 1. The auxiliary model 2. RS cavit...

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A Note on the Guerra and Talagrand Theorems for Mean Field Spin Glasses: The Simple Case of Spherical Models

Cited by 6 publications

References 24 publications

Legendre Duality of Spherical and Gaussian Spin Glasses

Legendre Duality of Spherical and Gaussian Spin Glasses

Effect of quantum fluctuations on the coloring of random graphs

Mean field theory of spin glasses

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