Parisi Replica Symmetry Breaking Research Articles

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 Sherrington–Kirkpatrick model near Tc and near T=0

Some recent results concerning the Sherrington–Kirkpatrick model are reported. For T near the critical temperature T c , the replica free energy of the Sherrington–Kirkpatrick model is taken as the starting point of an expansion in powers of about the replica symmetric solution . The expansion is kept up to fourth order in δ Q where a Parisi solution Q ab = Q(x) emerges, but only if one remains close enough to T c . For T near zero we show how to separate contributions from x ≪ T ≪ 1 where the Hessian maintains the standard structure of Parisi replica symmetry breaking with bands of eigenvalues bounded below by zero modes. For T ≪ x ≤ 1 the bands collapse and only two eigenvalues, a null one and a positive one, are found. In this region the solution stands in what can be called a droplet-like regime.

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.

Clusters of solutions and replica symmetry breaking in randomk-satisfiability

We study the set of solutions of randomk-satisfiability formulas through the cavity method. It is known that, for aninterval of the clause-to-variables ratio, this decomposes into an exponentialnumber of pure states (clusters). We refine substantially this picture by: (i)determining the precise location of the clustering transition; (ii) uncovering asecond ‘condensation’ phase transition in the structure of the solution set fork≥4. These results both follow from computing the large deviation rate of the internal entropyof pure states. From a technical point of view our main contributions are a simplified version ofthe cavity formalism for special values of the Parisi replica symmetry breaking parameterm (inparticular for m = 1 via a correspondence with the tree reconstruction problem) and newlarge-k expansions.

Construction and purpose of effective field theories for frustrated magnetic order

AbstractThis article reviews recent years' progress in the low temperature analysis of standard models of spin glass order such as the Sherrington‐Kirkpatrick (SK) model. Applications to CdTe/CdMnTe layered systems and explanation of glassy antiferromagnetic order at lowest temperatures stimulated us to study in detail the beautifully complex physical effects of replica symmetry breaking (RSB).We discuss analytical ideas based on highly precise numerical data which lead to the construction of relatively simple effective field theories for the SK model and help to understand the mysterious features of its exact solution. The goal is to find construction principles for the theory of interplay between frustrated magnetic order and various relevant physical degrees of freedom. The emphasis in this article is on the role of Parisi's RSB, which surprisingly creates critical phenomena in the low temperature limit despite the absence of a standard phase transition. (© 2007 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Survey propagation for the cascading Sourlas code

We investigate how insights from statistical physics, namely survey propagation, can improve decoding of a particular class of sparse error correcting codes. We show that a recently proposed algorithm, time averaged belief propagation, is in fact intimately linked to a specific survey propagation for which Parisi's replica symmetry breaking parameter is set to zero, and that the latter is always superior to belief propagation in the high connectivity limit. We briefly look at further improvements available by going to the second level of replica symmetry breaking.

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.

The stability of the replica-symmetric state in finite-dimensional spin glasses

According to the droplet picture of spin glasses, the low-temperature phase of spin glasses should be replica symmetric. However, analysis of the stability of this state suggested that it was unstable and this instability lends support to the Parisi replica symmetry breaking picture of spin glasses. The finite-size scaling functions in the critical region of spin glasses below T_c in dimensions greater than 6 can be determined and for them the replica symmetric solution is unstable order by order in perturbation theory. Nevertheless the exact solution can be shown to be replica-symmetric. It is suggested that a similar mechanism might apply in the low-temperature phase of spin glasses in less than six dimensions, but that a replica symmetry broken state might exist in more than six dimensions.

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.

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.

An exact solution for Parisi equations with R steps of RSB, Free energy and fluctuations

We show that there is no need to modify the Parisi replica symmetry breaking ansatz, by working with R steps of breaking and solving exactly the discrete stationarity equations generated by the standard 'truncated Hamiltonian' of spin glass theory.

Spin-glass effects on critical behavior in systems with extended quenched disorder

A renormalization-group approach is used for investigating the effects of the Parisi replica symmetry breaking (``spin-glass effects'') on critical properties of d-dimensional systems with extended quenched disorder along ${\ensuremath{\varepsilon}}_{d}$ directions. Within a double expansion scheme to first order in $(\ensuremath{\varepsilon}=4\ensuremath{-}d, {\ensuremath{\varepsilon}}_{d})$ it is found that for $d&lt;~4+{\ensuremath{\varepsilon}}_{d}$ and order-parameter dimensionalities $n&lt;~4,$ a full fixed-point instability with a consequent runaway towards a strong-coupling regime occurs in the parameter space while, for $d&lt;4$ and $n&gt;4,$ compatibly with the modified Harris criterion, an additional marginally stable fixed point exists which might control a new anisotropic random critical behavior. In any case, the traditional replica-symmetric Dorogovtsev-like solution for classical and quantum systems is unstable against the replica symmetry-breaking modes.

Evidence for the Droplet Picture of Spin Glasses

We have studied the Parisi overlap distribution for the three dimensional Ising spin glass in the Migdal-Kadanoff approximation. For temperatures T around 0.7Tc and system sizes upto L=32, we found a P(q) as expected for the full Parisi replica symmetry breaking, just as was also observed in recent Monte Carlo simulations on a cubic lattice. However, for lower temperatures our data agree with predictions from the droplet or scaling picture. The failure to see droplet model behaviour in Monte Carlo simulations is due to the fact that all existing simulations have been done at temperatures too close to the transition temperature so that sytem sizes larger than the correlation length have not been achieved.

THREE-DIMENSIONAL QUANTUM SPIN GLASS THEORY(Ⅲ) ──REPLICA SYMMETRY BREAKING SOLUTION

Using generalized Parisi's replica symmetry breaking scheme, the quantum spin glass theory proposed by our previous work is extended to the first-step replica symmetry breaking. A set of new self-consistent equations of spin self-interact ions and spin glass order parameters are obtained. The corresponding local susceptibility is reduced. It is found that the replica symmetry solution for vector spin glass model at low temperature must be broken through calculating the free-energy of the replica symmetry breaking.

ISING SPIN GLASS: RECENT PROGRESS IN THE FIELD THEORY APPROACH

A loop expansion around Parisi's replica symmetry breaking mean field theory is constructed, in zero field. We obtain the equation of state (and associated Parisi's solution) below the upper critical dimension du=6, and, in particular, explicit corrections in εlnt and (εlnt)2 with t=(Tc−T)/Tc and ε=6−d. This allows us to verify that standard scaling is satisfied with β=1+ε/2+0(ε2). We have also investigated the transverse (replicon) correlation function, particularly at zero overlap. Near du, G00 (p) ~ t/P4 (for p~0) is the most obvious obstacle to a meaningful theory in d=3. If we make the twofold assumption that (i) scaling applies, and (ii) the replicon propagator is dominated by the spectrum of the (small) transverse masses, we obtain the softer behavior G00(p) ~ (1/p2–η). (t/p1/v)β and a prediction for a soft replicon spectrum in tγk2γ/β γ/β instead of tk2 at du. We have checked tγ to one loop and work is in progress to check k2γ/β to the same order. Taking the above divergence in p−(d+2−η)/2 as the leading divergence defines the lower critical dimension d ℓ by d ℓ=2−η (d ℓ). Known values of η (at d=6, from the ε expansion at Tc, or from numerical work at d=4, 3) are compatible with d ℓ ~ 2.5±.3.

Short range corrections to the order parameter and to the excitation spectrum of the Ising spin glass

Abstract A consistent loop expansion around Parisi's replica symmetry breaking mean field theory for the Ising spin glass is constructed. Above the upper critical dimension d = 6 the short range corrections can be absorbed into the parameters of a Ginzburg-Landau type free energy functional whereby the d 8 the map is regular, thus mean field exponents are preserved, only prefactors are modified. For 6

On the thermodynamics of the spin-glass state in infinite dimension

Some exact results for the infinitely long-range Ising spin glass are derived at arbitrary temperature and magnetic field below the transition line within Parisi's replica symmetry breaking theory. The distribution of overlaps between pure states, P(q), proves to be non-analytic at q=0 in zero field. The usual relation S=- delta F/ delta T is found to break down, showing that the thermodynamic limit and the temperature derivative do not commute in the ordered phase, at least in non-zero magnetic field. The upper breakpoint of the order parameter function and the Edwards-Anderson order parameter are rigorously derived in the low-temperature limit. The coefficient of the leading low-temperature correction to the latter turns out to be different from that derived from the TAP equations, indicating that Parisi's theory and the TAP equations may not be strictly equivalent.

Replica symmetry breaking in finite connectivity systems: a large connectivity expansion at finite and zero temperature

Parisi replica symmetry breaking is extended to random bond systems with finite, fixed connectivity (M+1). The free energy, f is explicitly calculated for large M within the first stage of symmetry breaking. At finite temperature T, the 1/M expansion is found to diverge as T to O. Indeed, direct evaluation at zero temperature shows that the large M expansion is in powers of 1/M. Introduction of symmetry breaking brings the value of f close ( approximately 1% for 10<M<20) to numerical estimates of Banavar et al. The same techniques apply to systems with average finite connectivity.

Replica symmetry breaking in weak connectivity systems

The authors propose a generalisation of Parisi's replica symmetry breaking scheme (1979) to spin glasses (or optimisation problems) described by a set of order parameters Q(r)alpha 1. . . alpha r, or equivalently by a global order parameter G( sigma alpha ), rather than by Q(r)alpha 1 alpha 2 alone. They study the particular case of Derrida's p-spin model (1981) in the very dilute limit. The model is solved exactly in the large p limit and a freezing transition occurs. Using replicas and a single step of their replica symmetry breaking ansatz they recover the exact result. They discuss the implications for the global order parameter G( sigma alpha ) when more than one step in the replica symmetry breaking process is taken.

Application of statistical mechanics to combinatorial optimization problems: The chromatic number problem andq-partitioning of a graph

Methods of statistical mechanics are applied to two important NP-complete combinatorial optimization problems. The first is the chromatic number problem, which seeks the minimal number of colors necessary to color a graph such that no two sites connected by an edge have the same color. The second is partitioning of a graph intoq equal subgraphs so as to minimize intersubgraph connections. Both models are mapped into a frustrated Potts model, which is related to theq- state Potts spin glass. For the first problem, we obtain very good agreement with numerical simulations and theoretical bounds using the annealed approximation. The quenched model is also discussed. For the second problem we obtain analytic and numerical results by evaluating the groundstate energy of theq=3 and 4 Potts spin glass using Parisi's replica symmetry breaking. We also perform some numerical simulations to test the theoretical result and obtain very good agreement.