Recently, Michel Talagrand computed the large deviations limit $\lim_{N\to\infty}(Na)^{-1}\log \e Z_N^a$ for the moments of the partition function $Z_N$ in the Sherrington-Kirkpatrick model for all real $a\geq 0.$ For $a\geq 1$ the limit is given by Guerra's inverse bound and this result extends the classical physicist's replica method that corresponds to integer $a.$ We give a new proof for $a\geq 1$ in the case of the pure $p$-spin SK model that provides a strong exponential control of the overlap.
We develop a cavity method in the spherical Sherrington-Kirkpatrick model at high temperature and small external field. As one application we compute the limit of the covariance matrix for fluctuations of the overlap and magnetization.
We prove the existence of a critical regime for the fluctuations of the ground-state energy of the spherical Sherrington-Kirkpatrick model in an external field, confirming predictions given in [3,12]. We also establish a critical regime for the fluctuations in a model with a critical Ferromagnetic interaction term, producing a three-parameter family of distributions generalizing the two-parameter family given in [4]. These results are both established in the generality of a $β$-ensemble analogue of the spherical Sherrington-Kirkpatrick model, which subsumes the complex and quarternionic genera…
In this paper we consider central limit theorems for various macroscopic observables in the high temperature region of the Sherrington-Kirkpatrick spin glass model. With a particular focus on obtaining a quenched central limit theorem for the energy density of the system with non-zero external field, we show how to combine the mean field cavity method with Stein's method in the quenched regime. The result for the energy density extends the corresponding result of Comets-Neveu.
We consider the free energy of the bipartite spherical Sherrington--Kirkpatrick model. We find the critical temperature and prove the limiting free energy for all non-critical temperature. We also show that the law of the fluctuation of the free energy converges to the Gaussian distribution when the temperature is above the critical temperature, and to the GOE Tracy--Widom distribution when the temperature is below the critical temperature. The result is universal, and the analysis is applicable to a more general setting including the case where the disorders are non-identically distributed.
The spherical Sherrington-Kirkpatrick model is a spherical mean field model for spin glass. We consider the fluctuations of the free energy at arbitrary non-critical temperature for the 2-spin model with no magnetic field. We show that in the high temperature regime the law of the fluctuations converges to the Gaussian distribution just like in the Sherrington-Kirtkpatrick model. We show, on the other hand, that the law of the fluctuations is given by the GOE Tracy-Widom distribution in the low temperature regime. The orders of the fluctuations are markedly different in these two regimes. A un…
We explore the joint behavior of a finite number of multi-overlaps in the high temperature phase of the SK model. Extending work by M. Talagrand, we show that, when these objects are scaled to have non-trivial limiting distributions, the joint behavior is described by a Gaussian process with an explicit covariance structure.
We present a new dynamical proof of the Thouless-Anderson-Palmer (TAP) equations for the classical Sherrington-Kirkpatrick spin glass at sufficiently high temperature. In our derivation, the TAP equations are a simple consequence of the decay of the two point correlation functions. The methods can also be used to establish the decay of higher order correlation functions. We illustrate this by proving a suitable decay bound on the three point functions from which we derive an analogue of the TAP equations for the two point functions.
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.
We study the phase transition of the free energy in the Sherrington-Kirkpatrick mean-field spin glass model with centered Gaussian external field. We show that the corresponding Almeida-Thouless line is the correct transition curve that distinguishes between the replica symmetric and replica symmetry breaking solutions in the Parisi formula.
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].
We consider the quantum Sherrington-Kirkpatrick (SK) spin-glass model with transverse field and provide a formula for its free energy in the thermodynamic limit, valid for all inverse temperatures $β>0$. To characterize the free energy, we use the path integral representation of the partition function and approximate the model by a sequence of finite-dimensional vector-spin glasses with $\mathbb{R}^d$-valued spins. This enables us to use results of Panchenko who generalized in \cite{Pan2,Pan3} the Parisi formula to classical vector-spin glasses. As a consequence, we can express the thermodynam…
We study the universality of superconcentration for the free energy in the Sherrington-Kirkpatrick (SK) model. In arXiv:0907.3381, Chatterjee showed that when the system consists of $N$ spins and Gaussian disorders, the variance of this quantity is superconcentrated by establishing an upper bound of order $N/\log{N}$, in contrast to the $O(N)$ bound obtained from the Gaussian-Poincaré inequality. In this paper, we show that superconcentration indeed holds for any choice of centered disorders with finite third moment, where the upper bound is expressed in terms of an auxiliary nondecreasing fun…
The authors of [Ann. Henri Poincaré 16 (2015) 691-708] introduced a multi-species version of the Sherrington-Kirkpatrick model and suggested the analogue of the Parisi formula for the free energy. Using a variant of Guerra's replica symmetry breaking interpolation, they showed that, under certain assumption on the interactions, the formula gives an upper bound on the limit of the free energy. In this paper we prove that the bound is sharp. This is achieved by developing a new multi-species form of the Ghirlanda-Guerra identities and showing that they force the overlaps within species to be com…
In this paper we study the equilibrium statistical mechanical as well as the dynamical properties of a Sherrington and Kirkpatrick model in a multi-bath setting introduced in [4]. We show that the free energy per particle in the thermodynamical limit obeys a variational principle of Parisi type. The relation between the resulting order parameters is discussed.
We consider a spin system with pure two spin Sherrington-Kirkpatrick Hamiltonian with Curie-Weiss interaction. The model where the spins are spherically symmetric was considered by \citet{Baiklee16} and \citet{Baikleewu18} which shows a two dimensional phase transition with respect to temperature and the coupling constant. In this paper we prove a result analogous to \citet{Baiklee16} in the "paramagnetic regime" when the spins are i.i.d. Rademacher. We prove the free energy in this case is asymptotically Gaussian and can be approximated by a suitable linear spectral statistics. Unlike the sph…
The existence theorem for replica-symmetry breaking (RSB) in the transverse field Sherrington-Kirkpatrick (SK) model is extended to the model with a general random exchange interactions. The relation between the expectation value of the exchange interaction energy and the Duhamel correlation function of spin operators can be obtained by an approximate integration by parts for general random interactions. In addition to the Falk-Bruch inequality, these explicit evaluations enable us to prove that the variance of overlap between two replica spin operators does not vanish under sufficiently weak …
We study the Replica Symmetric region of general multi-species Sherrington-Kirkpatrick (MSK) Model and answer some of the questions raised in Ann.~Probab.~43~(2015), no.~6, 3494--3513, where the author proved the Parisi formula under \emph{positive-definite} assumption on the disorder covariance matrix $Δ^2$. First, we prove exponential overlap concentration at high temperature for both \indf~and \emph{positive-definite} $Δ^2$ MSK model. We also prove a central limit theorem for the free energy using overlap concentration. Furthermore, in the zero external field case, we use a quadratic coupli…
The interpolation techniques have become, in the past decades, a powerful approach to lighten several properties of spin glasses within a simple mathematical framework. Intrinsically, for their construction, these schemes were naturally implemented into the cavity field technique, or its variants as the stochastic stability or the random overlap structures. However the first and most famous approach to mean field statistical mechanics with quenched disorder is the replica trick. Among the models where these methods have been used (namely, dealing with frustration and complexity), probably the …
We extend two rigorous results of Aizenman, Lebowitz, and Ruelle in their pioneering paper of 1987 on the Sherrington-Kirkpatrick spin-glass model without external magnetic field to the quantum case with a "transverse field" of strength $b$. More precisely, if the Gaussian disorder is weak in the sense that its standard deviation $v>0$ is smaller than the temperature $1/β$, then the (random) free energy almost surely equals the annealed free energy in the macroscopic limit and there is no spin-glass phase for any $b/v\geq0$. The macroscopic annealed free energy (times $β$) turns out to be non-…
We study the Potts spin glass model, which generalizes the Sherrington-Kirkpatrick model to the case when spins take more than two values but their interactions are counted only if the spins are equal. We obtain the analogue of the Parisi variational formula for the free energy, with the order parameter now given by a monotone path in the set of positive-semidefinite matrices. The main idea of the paper is a novel synchronization mechanism for blocks of overlaps. This mechanism can be used to solve a more general version of the Sherrington-Kirkpatrick model with vector spins interacting throug…
We continue our presentation of mathematically rigorous results about the Sherrington-Kirkpatrick mean field spin glass model. Here we establish some properties of the distribution of overlaps between real replicas. They are in full agreement with the Parisi accepted picture of spontaneous replica symmetry breaking. As a byproduct, we show that the selfaveraging of the Edwards-Anderson fluctuating order parameter, with respect to the external quenched noise, implies that the overlap distribution is given by the Sherrington-Kirkpatrick replica symmetric Ansatz. This extends previous results of …
Gibbs' measures in the Sherrington-Kirkpatrick type models satisfy two asymptotic stability properties, the Aizenman-Contucci stochastic stability and the Ghirlanda-Guerra identities, which play a fundamental role in our current understanding of these models. In this paper we show that one can combine these two properties very naturally into one unified stability property.
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 th…
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…
