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Meanfield spin glass models from the cavityROSt perspective
 In Prospects in mathematical physics, volume 437 of Contemp. Math
, 2007
"... ABSTRACT. The SherringtonKirkpatrick spin glass model has been studied as a source of insight into the statistical mechanics of systems with highly diversified collections of competing low energy states. The goal of this summary is to present some of the ideas which have emerged in the mathematical ..."
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ABSTRACT. The SherringtonKirkpatrick spin glass model has been studied as a source of insight into the statistical mechanics of systems with highly diversified collections of competing low energy states. The goal of this summary is to present some of the ideas which have emerged in the mathematical study of its free energy. In particular, we highlight the perspective of the cavity dynamics, and the related variational principle. These are expressed in terms of Random Overlap Structures (ROSt), which are used to describe the possible states of the reservoir in the cavity step. The Parisi solution is presented as reflecting the ansatz that it suffices to restrict the variation to hierarchal structures which are discussed here in some detail. While the Parisi solution was proven to be correct, through recent works of F. Guerra and M. Talagrand, the reasons for the effectiveness of the Parisi ansatz still remain to be elucidated. We question whether this could be related to the quasistationarity of the special subclass of ROSts given by Ruelle’s hierarchal ‘random
The infinite volume limit in generalized mean field disordered models
 Markov Processes and Related Fields 9
, 2003
"... We generalize the strategy, we recently introduced to prove the existence of the thermodynamic limit for the SherringtonKirkpatrick and pspin models, to a wider class of mean field spin glass systems, including models with multicomponent and nonIsing type spins, mean field spin glasses with an a ..."
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We generalize the strategy, we recently introduced to prove the existence of the thermodynamic limit for the SherringtonKirkpatrick and pspin models, to a wider class of mean field spin glass systems, including models with multicomponent and nonIsing type spins, mean field spin glasses with an additional CurieWeiss interaction, and systems consisting of several replicas of the spin glass model, where replicas are coupled with terms depending on the mutual overlaps.
Thermodynamic Limit for MeanField Spin Models
, 311
"... If the BoltzmannGibbs state ωN of a meanfield Nparticle system with Hamiltonian HN verifies the condition ωN(HN) ≥ ωN(HN1 + HN2 for every decomposition N1 + N2 = N, then its free energy density increases with N. We prove such a condition for a wide class of spin models which includes the CurieW ..."
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If the BoltzmannGibbs state ωN of a meanfield Nparticle system with Hamiltonian HN verifies the condition ωN(HN) ≥ ωN(HN1 + HN2 for every decomposition N1 + N2 = N, then its free energy density increases with N. We prove such a condition for a wide class of spin models which includes the CurieWeiss model, its pspin generalizations (for both even and odd p), its random field version and also the finite pattern Hopfield model. For all these cases the existence of the thermodynamic limit by subadditivity and boundedness follows. 1 1
Variational Bounds for the Generalized Random Energy Model
, 2006
"... We compute the pressure of the random energy model (REM) and generalized random energy model (GREM) by establishing variational upper and lower bounds. For the upper bound, we generalize Guerra’s “broken replica symmetry bounds”, and identify the random probability cascade as the appropriate random ..."
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We compute the pressure of the random energy model (REM) and generalized random energy model (GREM) by establishing variational upper and lower bounds. For the upper bound, we generalize Guerra’s “broken replica symmetry bounds”, and identify the random probability cascade as the appropriate random overlap structure for the model. For the REM the lower bound is obtained, in the high temperature regime using Talagrand’s concentration of measure inequality, and in the low temperature regime using convexity and the high temperature formula. The lower bound for the GREM follows from the lower bound for the REM by induction. While the argument for the lower bound is fairly standard, our proof of the upper bound is new.
Energy landscape statistics of the random orthogonal model [arXiv:condmat/0207681
"... The Random Orthogonal Model (ROM) of MarinariParisiRitort [13, 14] is a model of statistical mechanics where the couplings among the spins are defined by a matrix chosen randomly within the orthogonal ensemble. It reproduces the most relevant properties of the Parisi solution of the SherringtonKi ..."
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The Random Orthogonal Model (ROM) of MarinariParisiRitort [13, 14] is a model of statistical mechanics where the couplings among the spins are defined by a matrix chosen randomly within the orthogonal ensemble. It reproduces the most relevant properties of the Parisi solution of the SherringtonKirkpatrick model. Here we compute the energy distribution, and work out an extimate for the twopoint correlation function. Moreover, we show exponential increase of the number of metastable states also for non zero magnetic field.
Dilution Robustness for Mean Field Ferromagnets
, 2008
"... In this work we compare two different random dilution of a mean field ferromagnet: the first model is built on a Bernoullidiluted network while the second lives on a Poissondiluted network. While it is known that the two models have in the thermodynamic limit the same free energy we investigate on ..."
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In this work we compare two different random dilution of a mean field ferromagnet: the first model is built on a Bernoullidiluted network while the second lives on a Poissondiluted network. While it is known that the two models have in the thermodynamic limit the same free energy we investigate on the structural constraints that the two models must fulfill. We rigorously derive for each model the set of identities for the multioverlaps distribution using different methods for the two dilutions: constraints in the former model are obtained by studying the consequences of the selfaveraging of the internal energy density, while in the latter are obtained by a stochasticstability technique. Finally we prove that the identities emerging in the two models are the same, showing robustness of the ferromagnetic properties of diluted networks with respect to the details of dilution.
A short course on mean field spin glasses
"... The main topic of this lecture series are diordered mean field spin systems. This first section will, however, be devoted to ordered spin systems, and more precisely essentially to the CurieWeiss model. This will be indispensable to appreciate later the much more complicated SherringtonKirkpatrick ..."
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The main topic of this lecture series are diordered mean field spin systems. This first section will, however, be devoted to ordered spin systems, and more precisely essentially to the CurieWeiss model. This will be indispensable to appreciate later the much more complicated SherringtonKirkpatrick spin glass. 1.1 Spin systems. In his Ph.D. thesis in 1924, Ernst Ising [18, 19] attempted to solve a model, proposed by his advisor Lenz, intended to describe the statistical mechanics of an interacting system of magnetic moments. The setup of the model proceeds again from a lattice, Zd, and a finite subset, Λ ⊂ Zd. The lattice is supposed to represent the positions of the atoms in a regular crystal. Each atom is endowed with a magnetic moment that is quantized and can take only the two values +1 and −1, called the spin of the atom. This spin variable at site x ∈ Λ is denoted by σx. The spins are supposed to interact via an interaction potential φ(x, y); in addition, a magnetic field h is present. The energy of a spin configuration is then