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Fabio Lucio Toninelli - One of the best experts on this subject based on the ideXlab platform.
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the high temperature region of the viana bray diluted spin glass model
Journal of Statistical Physics, 2004Co-Authors: Francesco Guerra, Fabio Lucio ToninelliAbstract:In this paper, we study the high temperature or low connectivity phase of the Viana–Bray model in the absence of magnetic field. This is a diluted version of the well known Sherrington–Kirkpatrick mean field spin glass. In the whole replica symmetric region, we obtain a complete control of the system, proving annealing for the infinite volume free energy and a central limit theorem for the suitably rescaled fluctuations of the multi-overlaps. Moreover, we show that free energy fluctuations, on the scale 1/N, converge in the infinite volume limit to a non-Gaussian random variable, whose variance diverges at the boundary of the replica-symmetric region. The connection with the fully connected Sherrington– Kirkpatrick model is discussed.
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the high temperature region of the viana bray diluted spin glass model
arXiv: Disordered Systems and Neural Networks, 2003Co-Authors: Francesco Guerra, Fabio Lucio ToninelliAbstract:In this paper, we study the high temperature or low connectivity phase of the Viana-Bray model. This is a diluted version of the well known Sherrington-Kirkpatrick mean field spin glass. In the whole replica symmetric region, we obtain a complete control of the system, proving annealing for the infinite volume free energy, and a central limit theorem for the suitably rescaled fluctuations of the multi-overlaps. Moreover, we show that free energy fluctuations, on the scale 1/N, converge in the infinite volume limit to a non-Gaussian random variable, whose variance diverges at the boundary of the replica-symmetric region. The connection with the fully connected Sherrington-Kirkpatrick model is discussed.
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about the almeida thouless transition line in the sherrington Kirkpatrick mean field spin glass model
EPL, 2002Co-Authors: Fabio Lucio ToninelliAbstract:We consider the Sherrington-Kirkpatrick model and we prove that the quenched free energy per spin is strictly larger than the corresponding replica symmetric approximation, for all values of the temperature and of the magnetic field below the Almeida-Thouless line. This implies the breaking of replica symmetry in the whole expected spin glass phase.
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quadratic replica coupling in the sherrington Kirkpatrick mean field spin glass model
Journal of Mathematical Physics, 2002Co-Authors: Francesco Guerra, Fabio Lucio ToninelliAbstract:We develop a very simple method to study the high temperature, or equivalently high external field, behavior of the Sherrington–Kirkpatrick mean field spin glass model. The basic idea is to couple two different replicas with a quadratic term, trying to push out the two replica overlap from its replica symmetric value. In the case of zero external field, our results reproduce the well known validity of the annealed approximation, up to the known critical value for the temperature. In the case of nontrivial external field, we can prove the validity of the Sherrington–Kirkpatrick replica symmetric solution up to a line, which falls short of the Almeida–Thouless line, associated to the onset of the spontaneous replica symmetry breaking, in the Parisi ansatz. The main difference with the method, recently developed by Michel Talagrand, is that we employ a quadratic coupling, and not a linear one. The resulting flow equations, with respect to the parameters of the model, turn out to be very simple, and the param...
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quadratic replica coupling in the sherrington Kirkpatrick mean field spin glass model
arXiv: Disordered Systems and Neural Networks, 2002Co-Authors: Francesco Guerra, Fabio Lucio ToninelliAbstract:We develop a simple method to study the high temperature, or high external field, behavior of the Sherrington-Kirkpatrick mean field spin glass model. The basic idea is to couple two different replicas with a quadratic term, trying to push out the two replica overlap from its replica symmetric value. In the case of zero external field, our results reproduce the well known validity of the annealed approximation, up to the known critical value for the temperature. In the case of nontrivial external field, we prove the validity of the Sherrington-Kirkpatrick replica symmetric solution up to a line, which falls short of the Almeida-Thouless line, associated to the onset of the spontaneous replica symmetry breaking, in the Parisi Ansatz. The main difference with the method, recently developed by Michel Talagrand, is that we employ a quadratic coupling, and not a linear one. The resulting flow equations, with respect to the parameters of the model, turn out to be much simpler, and more tractable. By applying the cavity method, we show also how to determine free energy and overlap fluctuations, in the region where replica symmetry has been shown to hold.
Francesco Guerra - One of the best experts on this subject based on the ideXlab platform.
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the replica symmetric region in the sherrington Kirkpatrick mean field spin glass model the almeida thouless line
arXiv: Disordered Systems and Neural Networks, 2006Co-Authors: Francesco GuerraAbstract:In previous work, we have developed a simple method to study the behavior of the Sherrington-Kirkpatrick mean field spin glass model for high temperatures, or equivalently for high external fields. The basic idea was to couple two different replicas with a quadratic term, trying to push out the two replica overlap from its replica symmetric value. In the case of zero external field, our results reproduced the well known validity of the annealed approximation, up to the known critical value for the temperature. In the case of nontrivial external field, our method could prove the validity of the Sherrington-Kirkpatrick replica symmetric solution up to a line, which fell short of the Almeida-Thouless line, associated to the onset of the spontaneous replica symmetry breaking, in the Parisi Ansatz. Here, we make a strategic improvement of the method, by modifying the flow equations, with respect to the parameters of the model. We exploit also previous results on the overlap fluctuations in the replica symmetric region. As a result, we give a simple proof that replica symmetry holds up to the critical Almeida-Thouless line, as expected on physical grounds. Our results are compared with the characterization of the replica symmetry breaking line previously given by Talagrand. We outline also a possible extension of our methods to the broken replica symmetry region.
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the high temperature region of the viana bray diluted spin glass model
Journal of Statistical Physics, 2004Co-Authors: Francesco Guerra, Fabio Lucio ToninelliAbstract:In this paper, we study the high temperature or low connectivity phase of the Viana–Bray model in the absence of magnetic field. This is a diluted version of the well known Sherrington–Kirkpatrick mean field spin glass. In the whole replica symmetric region, we obtain a complete control of the system, proving annealing for the infinite volume free energy and a central limit theorem for the suitably rescaled fluctuations of the multi-overlaps. Moreover, we show that free energy fluctuations, on the scale 1/N, converge in the infinite volume limit to a non-Gaussian random variable, whose variance diverges at the boundary of the replica-symmetric region. The connection with the fully connected Sherrington– Kirkpatrick model is discussed.
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the high temperature region of the viana bray diluted spin glass model
arXiv: Disordered Systems and Neural Networks, 2003Co-Authors: Francesco Guerra, Fabio Lucio ToninelliAbstract:In this paper, we study the high temperature or low connectivity phase of the Viana-Bray model. This is a diluted version of the well known Sherrington-Kirkpatrick mean field spin glass. In the whole replica symmetric region, we obtain a complete control of the system, proving annealing for the infinite volume free energy, and a central limit theorem for the suitably rescaled fluctuations of the multi-overlaps. Moreover, we show that free energy fluctuations, on the scale 1/N, converge in the infinite volume limit to a non-Gaussian random variable, whose variance diverges at the boundary of the replica-symmetric region. The connection with the fully connected Sherrington-Kirkpatrick model is discussed.
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quadratic replica coupling in the sherrington Kirkpatrick mean field spin glass model
Journal of Mathematical Physics, 2002Co-Authors: Francesco Guerra, Fabio Lucio ToninelliAbstract:We develop a very simple method to study the high temperature, or equivalently high external field, behavior of the Sherrington–Kirkpatrick mean field spin glass model. The basic idea is to couple two different replicas with a quadratic term, trying to push out the two replica overlap from its replica symmetric value. In the case of zero external field, our results reproduce the well known validity of the annealed approximation, up to the known critical value for the temperature. In the case of nontrivial external field, we can prove the validity of the Sherrington–Kirkpatrick replica symmetric solution up to a line, which falls short of the Almeida–Thouless line, associated to the onset of the spontaneous replica symmetry breaking, in the Parisi ansatz. The main difference with the method, recently developed by Michel Talagrand, is that we employ a quadratic coupling, and not a linear one. The resulting flow equations, with respect to the parameters of the model, turn out to be very simple, and the param...
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broken replica symmetry bounds in the mean field spin glass model
arXiv: Disordered Systems and Neural Networks, 2002Co-Authors: Francesco GuerraAbstract:By using a simple interpolation argument, in previous work we have proven the existence of the thermodynamic limit, for mean field disordered models, including the Sherrington-Kirkpatrick model, and the Derrida p-spin model. Here we extend this argument in order to compare the limiting free energy with the expression given by the Parisi Ansatz, and including full spontaneous replica symmetry breaking. Our main result is that the quenched average of the free energy is bounded from below by the value given in the Parisi Ansatz uniformly in the size of the system. Moreover, the difference between the two expressions is given in the form of a sum rule, extending our previous work on the comparison between the true free energy and its replica symmetric Sherrington-Kirkpatrick approximation. We give also a variational bound for the infinite volume limit of the ground state energy per site.
Dmitry Panchenko - One of the best experts on this subject based on the ideXlab platform.
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free energy in the mixed p spin models with vector spins
arXiv: Probability, 2015Co-Authors: Dmitry PanchenkoAbstract:Using the synchronization mechanism developed in the previous work on the Potts spin glass model, arXiv:1512.00370, we obtain the analogue of the Parisi formula for the free energy in the mixed even $p$-spin models with vector spins, which include the Sherrington-Kirkpatrick model with vector spins interacting through their scalar product. As a special case, this also establishes the sharpness of Talagrand's upper bound for the free energy of multiple mixed $p$-spin systems coupled by constraining their overlaps.
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the sherrington Kirkpatrick model
2013Co-Authors: Dmitry PanchenkoAbstract:Preface.- 1 The Free Energy and Gibbs Measure.- 2 The Ruelle Probability Cascades.- 3 The Parisi Formula.- 4 Toward a Generalized Parisi Ansatz.- A Appendix.- Bibliography.- Notes and Comments.- References.- Index.
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the sherrington Kirkpatrick model an overview
arXiv: Mathematical Physics, 2012Co-Authors: Dmitry PanchenkoAbstract:The goal of this paper is to review some of the main ideas that emerged from the attempts to confirm mathematically the predictions of the celebrated Parisi ansatz in the Sherrington-Kirkpatrick model. We try to focus on the big picture while sketching the proofs of only a few selected results, but an interested reader can find most of the missing details in [31] and [44].
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the sherrington Kirkpatrick model an overview
Journal of Statistical Physics, 2012Co-Authors: Dmitry PanchenkoAbstract:The goal of this paper is to review some of the main ideas that emerged from the attempts to confirm mathematically the predictions of the celebrated Parisi ansatz in the Sherrington-Kirkpatrick model. We try to focus on the big picture while sketching the proofs of only a few selected results, but an interested reader can find most of the missing details in Panchenko (The Sherrington-Kirkpatrick Model, Manuscript, 2012) and Talagrand (Mean-Field Models for Spin Glasses, Springer, Berlin, 2011).
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spin glass models from the point of view of spin distributions
arXiv: Probability, 2010Co-Authors: Dmitry PanchenkoAbstract:In many spin glass models, due to the symmetry among sites, any limiting joint distribution of spins under the annealed Gibbs measure admits the Aldous-Hoover representation encoded by a function $\sigma:[0,1]^4\to\{-1,+1\}$, and one can think of this function as a generic functional order parameter of the model. In a class of diluted models, and in the Sherrington-Kirkpatrick model, we introduce novel perturbations of the Hamiltonian that yield certain invariance and self-consistency equations for this generic functional order parameter and we use these invariance properties to obtain representations for the free energy in terms of $\sigma$. In the setting of the Sherrington-Kirkpatrick model, the self-consistency equations imply that the joint distribution of spins is determined by the joint distributions of the overlaps, and we give an explicit formula for $\sigma$ under the Parisi ultrametricity hypothesis. In addition, we discuss some connections with the Ghirlanda-Guerra identities and stochastic stability and describe the expected Parisi ansatz in the diluted models in terms of $\sigma$.
Neal Kirkpatrick - One of the best experts on this subject based on the ideXlab platform.
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tulsa criminal defense attorney neal Kirkpatrick
2014Co-Authors: Neal KirkpatrickAbstract:Tulsa Criminal Defense Attorney, Neal Kirkpatrick, handles Federal Crimes, Domestic Violence Charges, Drug Crimes, Firearms and Gun Charges, Military Crimes and Offenses, and Sex Crimes Charges
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federal firearms criminal defense attorney neal Kirkpatrick
2014Co-Authors: Neal KirkpatrickAbstract:Federal Firearms Criminal Defense Attorney | Neal Kirkpatrick Criminal Defense Lawyer assists with Military Crimes, Federal Crimes, Federal Appeals, Drug Charges, Firearms Charges, Gun Charges.
Erwin Bolthausen - One of the best experts on this subject based on the ideXlab platform.
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an iterative construction of solutions of the tap equations for the sherrington Kirkpatrick model
Communications in Mathematical Physics, 2014Co-Authors: Erwin BolthausenAbstract:We propose an iterative scheme for the solutions of the TAP-equations in the Sherrington–Kirkpatrick model which is shown to converge up to and including the de Almeida–Thouless line. The main tool is a representation of the iterations which reveals an interesting structure of them. This representation does not depend on the temperature parameter, but for temperatures below the de Almeida–Thouless line, it contains a part which does not converge to zero in the limit.
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an iterative construction of solutions of the tap equations for the sherrington Kirkpatrick model
arXiv: Probability, 2012Co-Authors: Erwin BolthausenAbstract:We propose an iterative construction of solutions of the Thouless-Anderson-Palmer-equations for the Sherrington-Kirpatrick model. The iterative scheme is proved to converge exactly up to the de Almayda-Thouless-line. No results on the SK-model itself are derived.
