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146
The Parisi formula
, 2006
"... Using Guerra’s interpolation scheme, we compute the free energy of the SherringtonKirkpatrick model for spin glasses at any temperature, confirming a celebrated prediction of G. Parisi. ..."
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Cited by 130 (4 self)
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Using Guerra’s interpolation scheme, we compute the free energy of the SherringtonKirkpatrick model for spin glasses at any temperature, confirming a celebrated prediction of G. Parisi.
On the Structure of QuasiStationary Competing Particles Systems
, 2007
"... We study point processes on the real line whose configurations X are locally finite, have a maximum, and evolve through increments which are functions of correlated gaussian variables. The correlations are intrinsic to the points and quantified by a matrix Q = {qij}i,j∈N. A probability measure on th ..."
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Cited by 42 (5 self)
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We study point processes on the real line whose configurations X are locally finite, have a maximum, and evolve through increments which are functions of correlated gaussian variables. The correlations are intrinsic to the points and quantified by a matrix Q = {qij}i,j∈N. A probability measure on the pair (X, Q) is said to be quasistationary if the joint law of the gaps of X and of Q is invariant under the evolution. A known class of universally quasistationary processes is given by the Ruelle Probability Cascades (RPC), which are based on hierarchally nested PoissonDirichlet processes. It was conjectured that up to some natural superpositions these processes exhausted the class of laws which are robustly quasistationary. The main result of this work is a proof of this conjecture for the case where qij assume only a finite number of values. The result is of relevance for meanfield spin glass models, where the evolution corresponds to the cavity dynamics, and where the hierarchal organization of the Gibbs measure was first proposed as an ansatz.
Tight bounds for LDPC and LDGM codes under map decoding
 IEEE Transactions on Information Theory
"... A new method for analyzing low density parity check (LDPC) codes and low density generator matrix (LDGM) codes under bit maximum a posteriori probability (MAP) decoding is introduced. The method is based on a rigorous approach to spin glasses developed by Francesco Guerra. It allows to construct low ..."
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Cited by 41 (3 self)
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A new method for analyzing low density parity check (LDPC) codes and low density generator matrix (LDGM) codes under bit maximum a posteriori probability (MAP) decoding is introduced. The method is based on a rigorous approach to spin glasses developed by Francesco Guerra. It allows to construct lower bounds on the entropy of the transmitted message conditional to the received one. Based on heuristic statistical mechanics calculations, we conjecture such bounds to be tight. The method is illustrated on codes with Poisson left degree distributions when used over binary input output symmetric channels. However, it is arguably much more general. UMR 8549, Unité Mixte de Recherche du Centre National de la Recherche Scientifique et de l ’ Ecole Normale
The high temperature region of the VianaBray diluted spin glass model
, 2008
"... In this paper, we study the high temperature or low connectivity phase of the VianaBray model. This is a diluted version of the well known SherringtonKirkpatrick mean field spin glass. In the whole replica symmetric region, we obtain a complete control of the system, proving annealing for the infi ..."
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Cited by 41 (2 self)
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In this paper, we study the high temperature or low connectivity phase of the VianaBray model. This is a diluted version of the well known SherringtonKirkpatrick 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 multioverlaps. Moreover, we show that free energy fluctuations, on the scale 1/N, converge in the infinite volume limit to a nonGaussian random variable, whose variance diverges at the boundary of the replicasymmetric region. The connection with the fully connected SherringtonKirkpatrick model is discussed.
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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Cited by 29 (1 self)
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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
Bounds for diluted meanfields spin glass models
, 2008
"... In an important recent paper, [2], S. Franz and M. Leone prove rigorous lower bounds for the free energy of the diluted pspin model and the Ksat model at any temperature. We show that the results for these two models are consequences of a single general principle. Our calculations are significantl ..."
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Cited by 28 (4 self)
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In an important recent paper, [2], S. Franz and M. Leone prove rigorous lower bounds for the free energy of the diluted pspin model and the Ksat model at any temperature. We show that the results for these two models are consequences of a single general principle. Our calculations are significantly simpler than those of [2], even in the replicasymmetric case.
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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Cited by 26 (2 self)
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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.
The mean field Ising model trough interpolating techniques
 J. Stat. Phys
, 2008
"... is to illustrate how some recent techniques developed within the framework of spin glasses do work on simpler model, focusing on the method and not on the analyzed system. To fulfill our will the candidate model turns out to be the paradigmatic mean field Ising model. The model is introduced and inv ..."
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Cited by 17 (12 self)
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is to illustrate how some recent techniques developed within the framework of spin glasses do work on simpler model, focusing on the method and not on the analyzed system. To fulfill our will the candidate model turns out to be the paradigmatic mean field Ising model. The model is introduced and investigated with the interpolation techniques. We show the existence of the thermodynamic limit, bounds for the free energy density, the explicit expression for the free energy with its suitable expansion via the order parameter, the selfconsistency relation, the phase transition, the critical behavior and the selfaveraging properties. At the end a formulation of a Parisilike theory is tried and discussed.
GriffithKellySherman correlation inequalities: a useful tool in the theory of error correcting codes
 IEEE Trans. Inform. Theory
, 2007
"... Abstract—It is shown that a correlation inequality of statistical mechanics can be applied to linear lowdensity paritycheck codes. Thanks to this tool we prove that, under a natural assumption, the exponential growth rate of regular lowdensity paritycheck (LDPC) codes, can be computed exactly by ..."
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Cited by 17 (12 self)
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Abstract—It is shown that a correlation inequality of statistical mechanics can be applied to linear lowdensity paritycheck codes. Thanks to this tool we prove that, under a natural assumption, the exponential growth rate of regular lowdensity paritycheck (LDPC) codes, can be computed exactly by iterative methods, at least on the interval where it is a concave function of the relative weight of code words. Then, considering communication over a binary input additive white Gaussian noise channel with a Poisson LDPC code we prove that, under a natural assumption, part of the GEXIT curve (associated to MAP decoding) can also be computed exactly by the belief propagation algorithm. The correlation inequality yields a sharp lower bound on the GEXIT curve. We also make an extension of the interpolation techniques that have recently led to rigorous results in spin glass theory and in the SAT problem. Index Terms—Correlation inequalities, density evolution, generalized EXIT (GEXIT) curve, growth rate, interpolation technique, iterative decoding, lowdensity paritycheck (LDPC) codes, spin glasses. I.