Replica Symmetry Breaking Scheme Research Articles

Parisi’s Theory of Spin Glass

The spin-glass phase is characterized by the existence of many pure states due to random exchange interactions between spins. Parisi established the novel concept of replica symmetry breaking (RSB) from Sherrington Kirkpatrick’s mean-field theory via an abstract replica trick. In this article, his RSB scheme is reviewed from the view point of infinitely many pure states.

A Spin Glass Model for Reconstructing Nonlinearly Encrypted Signals Corrupted by Noise

We define a (symmetric key) encryption of a signal mathbf{s}in {mathbb {R^N}} as a random mapping mathbf{s}mapsto mathbf y =(y_1,ldots ,y_M)^Tin mathbb {R}^M known both to the sender and a recipient. In general the recipients may have access only to images mathbf{y} corrupted by an additive noise of unknown strength. Given the encryption redundancy parameter (ERP) mu =M/Nge 1 and the signal strength parameter R=sqrt{sum _i {s_i^2/N}}, we consider the problem of reconstructing mathbf{s} from its corrupted image by a Least Square Scheme for a certain class of random Gaussian mappings. The problem is equivalent to finding the configuration of minimal energy in a certain version of spherical spin glass model, with squared Gaussian random interaction potential. We use the Parisi replica symmetry breaking scheme to evaluate the mean overlap p_{infty }in [0,1] between the original signal and its recovered image (known as ’estimate’) as Nrightarrow infty , for a given (’bare’) noise-to-signal ratio (NSR) gamma ge 0. Such an overlap is a measure of the quality of the signal reconstruction. We explicitly analyze the general case of linear-quadratic family of random mappings and discuss the full p_{infty } (gamma ) curve. When nonlinearity exceeds a certain threshold but redundancy is not yet too big, the replica symmetric solution is necessarily broken in some interval of NSR. We show that encryptions with a nonvanishing linear component permit reconstructions with p_{infty }>0 for any mu >1 and any gamma <infty , with p_{infty }sim gamma ^{-1/2} as gamma rightarrow infty . In contrast, for the case of purely quadratic nonlinearity, for any ERP mu >1 there exists a threshold NSR value gamma _c(mu ) such that p_{infty }=0 for gamma >gamma _c(mu ) making the reconstruction impossible. The behaviour close to the threshold is given by p_{infty }sim (gamma _c-gamma )^{3/4} and is controlled by the replica symmetry breaking mechanism.

The Full Replica Symmetry Breaking in the Ising Spin Glass on Random Regular Graph

In this paper, we extend the full replica symmetry breaking scheme to the Ising spin glass on a random regular graph. We propose a new martingale approach, that overcomes the limits of the Parisi-M\'ezard cavity method, providing a well-defined formulation of the full replica symmetry breaking problem in random regular graphs. Finally, we define the order parameters of the system and get a set of self-consistency equations for the order parameters and the free energy. We face up the problem only from a technical point of view: the physical meaning of this approach and the quantitative evaluation of the solution of the self-consistency equations will be discussed in next works.

Disorder Chaos in the Spherical Mean-Field Model

We consider the problem of disorder chaos in the spherical mean-field model. It is concerned about the behavior of the overlap between two independently sampled spin configurations from two Gibbs measures with the same external parameters. The prediction states that if the disorders in the Hamiltonians are slightly decoupled, then the overlap will be concentrated near a constant value. Following Guerra's replica symmetry breaking scheme, we establish this at the level of the free energy as well as the Gibbs measure irrespective the presence or absence of the external field.

Stability of the Parisi solution for the Sherrington–Kirkpatrick model near T = 0

To test the stability of the Parisi solution near T = 0, we study the spectrum of the Hessian of the Sherrington–Kirkpatrick model near T = 0, whose eigenvalues are the masses of the bare propagators in the expansion around the mean-field solution. In the limit T ≪ 1, two regions can be identified. In the first region, for x close to 0, where x is the Parisi replica symmetry breaking scheme parameter, the spectrum of the Hessian is not trivial and maintains the structure of the full replica symmetry breaking state found at higher temperatures. In the second region T ≪ x ⩽ 1, as T → 0, the components of the Hessian become insensitive to changes of the overlaps and the bands typical of the full replica symmetry breaking state collapse. In this region only two eigenvalues are found: a null one and a positive one, ensuring stability for T ≪ 1. In the limit T → 0, the width of the first region shrinks to 0 and only the positive and null eigenvalues survive. As byproduct we enlighten the close analogy between the static Parisi replica symmetry breaking scheme and the multiple time-scales approach of dynamics, and compute the static susceptibility showing that it equals the static limit of the dynamic susceptibility computed via the modified fluctuation dissipation theorem.

An extended ensemble Monte Carlo study of a lattice glass model

The thermodynamic transition to a glass phase from a liquid phase is studied in a simple lattice glass model both on a random graph and in two dimensions by employing an extended ensemble Monte Carlo method, which enables us to explore its equilibrium behavior in a high density regime. The results for the random-graph case support the scenario of an equilibrium glass transition as predicted from one-step replica symmetry breaking schemes via the cavity method. Numerical evidence for the glass transition is also found for the system in two dimensions. The equation of state measured indicates a discontinuous change in the compressibility at the transition point and the overlap distribution function has a sharp double peak structure in the glassy phase. They suggest that this singularity originates from a one-step replica symmetry breaking.

Inverse freezing in the Hopfield fermionic Ising spin glass

In this work it is studied the Hopfield fermionic spin glass model which allows interpolating from trivial randomness to a highly frustrated regime. Therefore, it is possible to investigate whether or not frustration is an essential ingredient which would allow this magnetic disordered model to present naturally inverse freezing by comparing the two limits, trivial randomness and highly frustrated regime and how different levels of frustration could affect such unconventional phase transition. The problem is expressed in the path integral formalism where the spin operators are represented by bilinear combinations of Grassmann variables. The Grand Canonical Potential is obtained within the static approximation and one-step replica symmetry breaking scheme. As a result, phase diagrams temperature {\it versus} the chemical potential are obtained for several levels of frustration. Particularly, when the level of frustration is diminished, the reentrance related to the inverse freezing is gradually suppressed.

Weight space structure and analysis using a finite replica number in the Isingperceptron

The weight space of the Ising perceptron, in which a set of random patterns isstored, is examined using the generating function of the partition functionϕ(n) = (1/N)log[Zn] as the dimension ofthe weight vector N tends to infinity, where Z is the partition function and represents the configurational average. We utilizeϕ(n) for two purposes, depending on the value of the ratioα = M/N, whereM is the number ofrandom patterns. For α<αs = 0.833..., we employ ϕ(n), in conjunction with Parisi’s one-step replica symmetry breaking scheme in the limit of , to evaluate the complexity that characterizes the number of disjoint clusters of weightsthat are compatible with a given set of random patterns, which indicates that, intypical cases, the weight space is equally dominated by a single large cluster ofexponentially many weights and exponentially many small clusters of a single weight. Forα>αs, on theother hand, ϕ(n) is used to assess the rate function of a small probability that a given set of random patterns is atypicallyseparable by the Ising perceptrons. We show that the analyticity of the rate function changes atα = αGD = 1.245..., which implies that the dominant configuration of the atypically separable patternsexhibits a phase transition at this critical ratio. Extensive numerical experiments areconducted to support the theoretical predictions.

Replica-symmetry breaking: discrete and continuous schemes in the Sherrington–Kirkpatrick model

We study hierarchies of replica-symmetry-breaking solutions of the Sherrington–Kirkpatrick model. Stationarity equations for order parameters of solutions with an arbitrary number of hierarchies are set and the limit to infinite number of hierarchical levels is discussed. In particular, we demonstrate how the continuous replica-symmetry breaking scheme of Parisi emerges and how the limit to infinite-many hierarchies leads to equations for the order-parameter function of the continuous solution. The general analysis is accompanied by an explicit asymptotic solution near the de Almeida–Thouless instability line in the nonzero magnetic field.

Marginal states in mean-field glasses

We study mean field systems whose free energy landscape is dominated by marginally stable states. We review and develop various techniques to describe such states, elucidating their physical meaning and the interrelation between them. In particular, we give a physical interpretation of the two-group replica symmetry breaking scheme and confirm it by establishing the relation to the cavity method and to the counting of solutions of the Thouless-Anderson-Palmer equations. We show how these methods all incorporate the presence of a soft mode in the free energy landscape and interpret the occurring order parameter functions in terms of correlations between the soft mode and the local magnetizations. The general formalism is applied to the prototypical case of the Sherrington-Kirkpatrick-model where we re-examine the physical properties of marginal states under a new perspective.

Low-Temperature Solution of the Sherrington-Kirkpatrick Model

I propose a simple scaling ansatz for the full replica symmetry breaking solution of the Sherrington-Kirkpatrick model in the low energy sector. This solution is shown to become exact in the limit x --> 0, Bx --> infinity of the Parisi replica symmetry breaking scheme parameter . The distribution function of the frozen fields has been known to develop a linear gap at zero temperature. The scaling equations are integrated to find an exact numerical value for the slope of the gap thetaP(x,y)/delta|(y --> 0) = 0.301 046.... I also use the scaling solution to devise an inexpensive numerical procedure for computing finite time scale (x =1) quantities. The entropy, the zero field cooled susceptibility, and the local field distribution function are computed in the low-temperature limit with high precision, barely achievable by currently available methods.

Replica symmetry breaking in the ‘small world’ spin glass

We apply the cavity method to an Ising spin glass model on a ‘small world’ lattice, a randombond graph superimposed upon a one-dimensional ferromagnetic ring, which allows us toidentify the cavity fields in the chain with eigenvectors of the replicated transfer matrix forthe chain using the one-step replica symmetry breaking scheme developed by Mézard andParisi for the Bethe lattice; we evaluate observables for a model with fixed connectivity and ± J long range bonds. We observe that ferromagnetic ordering within this model extends into alarger parameter region than predicted from replica symmetric calculations. Numericalsimulations, although suffering non-equilibration effects, seem to provide support for thisresult.

On a dynamical-like replica-symmetry-breaking scheme for the spin glass

Considering the unphysical result obtained in the calculation of the free-energy cost for twisting the boundary conditions in a spin glass, we trace it to the negative multiplicities associated with the Parisi replica-symmetry breaking (RSB). We point out that a distinct RSB, that keeps positive multiplicities, was proposed long ago, in the spirit of an ultra-long time dynamical approach due to Sompolinsky. For an homogeneous bulk system, both RSB schemes are known to yield identical free energies and observables. However, using the dynamical RSB, we have recalculated the twist free energy at the mean-field level. The free-energy cost of this twist is, as expected, positive in that scheme, as it should be.

Infinite Range Interaction Model of a Structural Glass

In this paper a simple mean-field model for the liquid-glass phase transition is proposed. This is the low density D-dimensional system of N particles interacting via infinite-range oscillating potential. In the framework of the replica approach it is shown that such a system exhibits the phase transition between the high-temperature liquid phase and the low-temperature glass phase. This phase transition is described in terms of the standard one-step replica symmetry breaking scheme.

CONVEX REPLICA SYMMETRY BREAKING FROM POSITIVITY AND THERMODYNAMIC LIMIT

Consider a correlated Gaussian random energy model built by successively adding one particle (spin) into the system and imposing the positivity of the associated covariance matrix. We show that the validity of a recently isolated condition ensuring the existence of the thermodynamic limit forces the covariance matrix to exhibit the Parisi replica symmetry breaking scheme with a convexity conditions on the matrix elements.

One step RSB scheme for the rate distortion function

We apply statistical mechanics to an inverse problem of linear mapping to investigate the physics of the irreversible compression. We use the replica symmetry breaking (RSB) technique with a toy model to demonstrate the Shannon result. The rate distortion function, which is widely known as the theoretical limit of the compression with a fidelity criterion, is derived using the Parisi one step RSB scheme. The bound cannot be achieved in the sparsely-connected systems, where suboptimal solutions dominate the capacity.

Rigorous decimation-based construction of ground pure states for spin-glass models on random lattices.

A constructive scheme for determining pure states at very low temperature in the 3-spins glass model on a random lattice is provided, in full agreement with Parisi's one step replica symmetry breaking (RSB) scheme. Proof is based on the analysis of a partial decimation procedure and of the statistical properties of its output, i.e., a reduced Hamiltonian acting on a subset of the initial spins. The number of ground states (GS) in each state, the number of states, and the distances between GS are calculated and correspond to RSB predictions.

Modulated replica symmetry breaking schemes for antiferrimagnetic spin glasses

We define modulated replica symmetry breaking (RSB) schemes which combine tree- and wave- like structures. A modulated scheme and unmodulated RSB are evaluated at 1-step level for a semicon- ductor model with antiferromagnetic Korenblit-Shender interaction. By comparison of the free energies we find evidence that a T = 0 phase transition in the ferrimagnetic phase leads to a transition between the different RSB-schemes. An embedding factor of Parisi block matrices with sublattice-asymmetrical size is employed as a new variational parameter in the modulated scheme. PACS. 68.35.Rh Phase transitions and critical phenomena - 75.10.Nr Spin glass and other random models

Spin glasses on Bethe lattices for large coordination number

We study spin glasses on random lattices with finite connectivity. In the infinite connectivity limit they reduce to the Sherrington Kirkpatrick model. In this paper we investigate the expansion around the high connectivity limit. Within the replica symmetry breaking scheme at two steps, we compute the free energy at the first order in the expansion in inverse powers of the average connectivity (z), both for the fixed connectivity and for the fluctuating connectivity random lattices. It is well known that the coefficient of the 1/z correction for the free energy is divergent at low temperatures if computed in the one step approximation. We find that this annoying divergence becomes much smaller if computed in the framework of the more accurate two steps breaking. Comparing the temperature dependance of the coefficients of this divergence in the replica symmetric, one step and two steps replica symmetry breaking, we conclude that this divergence is an artefact due to the use of a finite number of steps of replica symmetry breaking. The 1/z expansion is well defined also in the zero temperature limit.

Hierarchical self-programming in recurrent neural networks

We study self-programming in recurrent neural networks where both neurons (the 'processors') and synaptic interactions ('the programme') evolve in time simultaneously, according to specific coupled stochastic equations. The interactions are divided into a hierarchy of L groups with adiabatically separated and monotonically increasing time-scales, representing sub-routines of the system programme of decreasing volatility. We solve this model in equilibrium, assuming ergodicity at every level, and find as our replica-symmetric solution a formalism with a structure similar but not identical to Parisi's L-step replica symmetry breaking scheme. Apart from differences in details of the equations (due to the fact that here interactions, rather than spins, are grouped into clusters with different time-scales), in the present model the block sizes mi of the emerging ultrametric solution are not restricted to the interval [0, 1], but are independent control parameters, defined in terms of the noise strengths of the various levels in the hierarchy, which can take any value in [0, ∞⟩. This is shown to lead to extremely rich phase diagrams, with an abundance of first-order transitions especially when the level of stochasticity in the interaction dynamics is chosen to be low.