One-step Replica Symmetry Breaking Solution Research Articles

Next nearest neighbour Ising models on random graphs

This paper develops results for the next nearest neighbour Ising model on random graphs. Besides being an essential ingredient in classic models for frustrated systems, second neighbour interactions arise naturally in several applications, such as the colour diversity problem and graphical games. We demonstrate ensembles of random graphs, including regular connectivity graphs, that have a periodic variation of free energy, with either the ratio of nearest to next nearest couplings, or the mean number of nearest neighbours. When the coupling ratio is integer then paramagnetic phases can be found at zero temperature. This is shown to be related to the locked or unlocked nature of the interactions. For anti-ferromagnetic couplings, spin glass phases are demonstrated at low temperature. The interaction structure is formulated as a factor graph, the solution on a tree is developed. The replica symmetric and energetic one-step replica symmetry breaking solution is developed using the cavity method. We calculate within these frameworks the phase diagram and demonstrate the existence of dynamical transitions at zero temperature for cases of anti-ferromagnetic coupling on regular and inhomogeneous random graphs.

One-step replica symmetry breaking solution for a highly asymmetric two-sublattice fermionic Ising spin-glass model in a transverse field

The one-step replica symmetry breaking (RSB) is used to study a two-sublattice fermionic infinite-range Ising spin glass (SG) model in a transverse field $\Gamma$. The problem is formulated in a Grassmann path integral formalism within the static approximation. In this model, a parallel magnetic field $H$ breaks the symmetry of the sublattices. It destroys the antiferromagnetic (AF) order, but it can favor the nonergodic mixed phase (SG+AF) characterizing an asymmetric RSB region. In this region, intra-sublattice disordered interactions $V$ increase the difference between the RSB solutions of each sublattice. The freezing temperature shows a higher increase with $H$ when $V$ enhances. A discontinue phase transition from the replica symmetry (RS) solution to the RSB solution can appear with the presence of an intra-sublattice ferromagnetic average coupling. The $\Gamma$ field introduces a quantum spin flip mechanism that suppresses the magnetic orders leading them to quantum critical points. Results suggest that the quantum effects are not able to restore the RS solution. However, in the asymmetric RSB region, $\Gamma$ can produce a stable RS solution at any finite temperature for a particular sublattice while the other sublattice still presents RSB solution for the special case in which only the intra-sublattice spins couple with disordered interactions.

Statistical Mechanical Study of Code-Division Multiple-Access Multiuser Detectors –Analysis of Replica Symmetric and One-Step Replica Symmetry Breaking Solutions–

We study the problem of performance evaluation of code-division multiple-access (CDMA) multiuser detectors by means of statistical mechanics using the replica method. The replica symmetric solutions were analyzed previously. As is well known, the replica symmetric solution becomes unstable when the temperature (magnitude of noise) is lowered. In this paper, we investigate both the behavior and the stability of the replica symmetric solutions in the low temperature region. We find that the solutions have complicated bifurcation structures in the low temperature region where the solutions coexist. We also find that there are two types of replica symmetry breaking, Almeida–Thouless (AT)-instability and freezing. We obtain the one-step replica symmetry breaking solution in each case. Further, we compare the theoretical results with the results from the Monte-Carlo simulations. Consequently, we find that the theoretical results agree with the numerical simulations.

One-step replica symmetry breaking solution of the quadrupolar glass model

We consider the quadrupolar glass model with infinite-range random interaction. Introducing a simple one-step replica symmetry breaking ansatz we investigate the para-glass continuous (discontinuous) transition which occurs below (above) a critical value of the quadrupole dimension m*. By using a mean-field approximation we study the stability of the one-step replica symmetry breaking solution and show that for m>m* there are two transitions. The thermodynamic transition is discontinuous but there is no latent heat. At a higher temperature we find the dynamical or glass transition temperature and the corresponding discontinuous jump of the order parameter.

One-step replica symmetry breaking solution for a fermionic Ising spin glass in a transverse field

The fermionic Ising spin glass models in a transverse field are investigated in a Grassmann path integral formalism. The Parisi's scheme of one-step replica symmetry breaking (RSB) is used within the static ansatz. This formalism has already been applied in a theory in which $m$ (Parisi's block-size parameter) is taken as a constant (PRB 72, 014447 (2005)). Now, it is extended to consider $m$ as a variational parameter. In this case, the results show that RSB is present when $T\to0$, in which the system is driven by quantum fluctuations.

Replica field theory for a polymer in random media

In this paper we revisit the problem of a (non-self-avoiding) polymer chain in a random medium which was previously investigated by Edwards and Muthukumar (EM) [J. Chem. Phys. 89, 2435 (1988)]. As noticed by Cates and Ball (CB) [J. Phys. (France) 49, 2009 (1988)] there is a discrepancy between the predictions of the replica calculation of EM and the expectation that in an infinite medium the quenched and annealed results should coincide (for a chain that is free to move) and a long polymer should always collapse. CB argued that only in a finite volume one might see a "localization transition" (or crossover) from a stretched to a collapsed chain in three spatial dimensions. Here we carry out the replica calculation in the presence of an additional confining harmonic potential that mimics the effect of a finite volume. Using a variational scheme with five variational parameters we derive analytically for d<4 the result R approximately (g|ln &mgr;|)(-1/(4-d)) approximately (g ln V)(-1/(4-d)), where R is the radius of gyration, g is the strength of the disorder, &mgr; is the spring constant associated with the confining potential, and V is the associated effective volume of the system. Thus the EM result is recovered with their constant replaced by ln V as argued by CB. We see that in the strict infinite volume limit the polymer always collapses, but for finite volume a transition from a stretched to a collapsed form might be observed as a function of the strength of the disorder. For d<2 and for large V>V' approximately exp(g(2/(2-d))L((4-d)/(2-d))) the annealed results are recovered and R approximately (Lg)(1/(d-2)), where L is the length of the polymer. Hence the polymer also collapses in the large L limit. The one-step replica symmetry breaking solution is crucial for obtaining the above results.

Analysis of Learning from Stochastic Rules in the Framework of Replica Symmetry Breaking

We study learning from examples by a perceptron when a return to an example is given by a stochastic relation which is represented by the conditional probability distribution P ( u ), where u is proportional to the inner product between the optimal synaptic weight vector and the example vector. The problem is analyzed by replica method in the case of spherical weights. Since the replica symmetric(RS) solution turns out to be unstable for stochastic cases, we consider the one-step replica symmetry breaking (RSB) solution. We investigate the asymptotic behavior of the learning curve as \(\alpha \equiv \frac{p}{N} \rightarrow \infty\), where p is the number of samples and N is the dimension of the synaptic weights. The average generalization error e g is expressed as (e g - e min ) ∝α - γ in the asymptotic region. For the minimum-error algorithm, we find \(\gamma=\frac{1+\delta}{1+3\delta}\) for the RS solution and \(\gamma=\frac{1+\delta}{1+2\delta}\) for the one-step RSB solution in the case of the functio...

Maximal stability in unsupervised learning

A perceptron can classify a set of random patterns into two groups. The maximal gap between the two groups is calculated using replica theory. The replica-symmetric solution is shown to be unstable and gives results which are qualitatively different from the one-step replica-symmetry breaking solution (RSB1). The results of a calculation without the replica method are given, yielding an exact upper bound for the gap which almost coincides with the RSB1 solution. Finding the classification of a set of patterns with a maximal gap is a difficult combinatorial optimization problem. An algorithm is developed which gives large gaps.

Replica symmetry breaking in neural networks with modified pseudo-inverse interactions

Replica symmetry breaking is studied in fully connected neural networks with modified pseudo-inverse interactions. The interaction matrix has an intermediate form between the Hebb learning rule and the pseudo-inverse one. At low temperatures there is a region of parameters where the replica symmetric solution is stable while its entropy is negative. It indicates the existence of an alternative solution in which the replica symmetry is broken. The one-step replica symmetry-breaking solution is found and its properties are analysed.