One-step Replica Symmetry Breaking Ansatz Research Articles

Dynamical arrest with zero complexity: The unusual behavior of the spherical Blume-Emery-Griffiths disordered model.

The short- and long-time dynamics of model systems undergoing a glass transition with apparent inversion of Kauzmann and dynamical arrest glass transition lines is investigated. These models belong to the class of the spherical mean-field approximation of a spin-1 model with p-body quenched disordered interaction, with p>2, termed spherical Blume-Emery-Griffiths models. Depending on temperature and chemical potential the system is found in a paramagnetic or in a glassy phase and the transition between these phases can be of a different nature. In specific regions of the phase diagram coexistence of low-density and high-density paramagnets can occur, as well as the coexistence of spin-glass and paramagnetic phases. The exact static solution for the glassy phase is known to be obtained by the one-step replica symmetry breaking ansatz. Different scenarios arise for both the dynamic and the thermodynamic transitions. These include: (i) the usual random first-order transition (Kauzmann-like) for mean-field glasses preceded by a dynamic transition, (ii) a thermodynamic first-order transition with phase coexistence and latent heat, and (iii) a regime of apparent inversion of static transition line and dynamic transition lines, the latter defined as a nonzero complexity line. The latter inversion, though, turns out to be preceded by a dynamical arrest line at higher temperature. Crossover between different regimes is analyzed by solving mode-coupling-theory equations near the boundaries of paramagnetic solutions and the relationship with the underlying statics is discussed.

Spin-glass splitting in the quantum Ghatak-Sherrington model

We propose an expanded spin-glass model, called the quantum Ghatak-Sherrington model, which considers spin-1 quantum spin operators in a crystal field and in a transverse field. The analytic solutions and phase diagrams of this model are obtained by using the one-step replica symmetry-breaking ansatz under the static approximation. Our results represent the splitting within one spin-glass (SG) phase depending on the values of crystal and transverse fields. The two separated SG phases, characterized by a density of filled states, show certain differences in their shapes and phase boundaries. Such SG splitting becomes more distinctive when the degeneracy of the empty states of spins is larger than one of their filled states.

Random, thermodynamic and inverse first-order transitions in the Blume–Capel spinglass

A class of model systems undergoing a glass transition with inversion of the fluid and glassyphase in temperature is investigated in order to qualitatively characterize the so-calledinverse freezing phenomenon occurring in some complex glassy polymeric systems such asmethylcellulose. The leading model we analyze is the spherical mean-field approximation of a spin-1 modelwith p-body quenched disordered interaction. Depending on temperature and chemical potentialthe system is found in a paramagnetic or in a glassy phase and the transition between thesephases can be of a different nature. In the given conditions inverse freezing occurs. Asp = 2 theglassy phase is replica-symmetric and the transition is always continuous in the phase diagram. Forp > 2 the exact solution for the glassy phase is obtained by the one-step replica symmetrybreaking ansatz. Different scenarios arise for both the dynamic and the thermodynamictransitions. These include (i) the usual random first-order transition (Kauzmann-like)preceded by a dynamic transition, typical of mean-field glasses, (ii) a thermodynamicfirst-order transition with phase coexistence and latent heat and (iii) a regime of inversionof static and dynamic transition lines. In the latter case a thermodynamic stable glassyphase, with zero configurational entropy, is dynamically accessible from the paramagneticphase. Crossover between different transition regimes is analyzed by means of replicasymmetry breaking theory and a detailed study of the complexity and of the stability ofthe static solution is performed throughout the space of external thermodynamicparameters.

The replica mean field theory for the glass matrix model

We study the dynamical phase transition of the glass matrix model by using the replica method for deterministic models. In the glass matrix model, N dynamical variables are defined by P component vectors which make states of a particle. Each component of a vector is assumed to take ±1. To perform replica calculation, auxiliary variables are introduced to control the order of the sum of replicated partition function. These variables work like quenched random variables of the usual replica method. Using the approximation for small P/N and assuming the homogeneity of the component space, we find that the replica theory is similar to that of the previously studied glassy spin models. We study the dynamical solution, which is defined by imposing the marginally stable condition for a one-step replica symmetry breaking ansatz. To find the marginally stable condition, we study the fluctuation modes in replica spaces and component space. The results of simulated annealing is presented to compare with the analytic result. The transition temperatures suggested by the two approaches are consistent. However, the agreement is modest in the sense that simulated annealing shows richer behavior than the replica results, mainly due to the lower energy states, which appear by the slow annealing schedule.

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.

How Many Colors to Color a Random Graph? Cavity, Complexity, Stability and All That

We review recent progress on the statiscal physics study of the problem of coloring random graphs with q colors. We discuss the existence of a threeshold at connectivity c_q=2q log q-log q-1+o(1) separting two phases which are respectivily COL(orable) and UNCOL(orable) with q colors; We also argue that the so-called one-step replica symmetry breaking ansatz used to derive these results give it exact threshold values, and draw a general phase diagram of the problem.

Structural glass on a lattice in the limit of infinite dimensions

We construct a mean-field theory for the lattice model of a structural glass and solve it using the replica method and one-step replica symmetry breaking ansatz; this theory becomes exact in the limit of infinite dimensions. Analyzing the stability of this solution we conclude that the metastable states remain uncorrelated in a finite temperature range below the transition, but become correlated at a certain low temperature; below this temperature metastable states bifurcate and the one step replica symmetry breaking solution fails. We find dynamic and thermodynamic transition temperatures as functions of the density, and construct a full thermodynamic description of a typical physical process in which the system is trapped in one metastable state when cooled below the vitrification temperature. We find that for such physical process the entropy and pressure at the glass transition are continuous across the transition, while their temperature derivatives have jumps.

On the Conditions for the Existence of Perfect Learning and Power Law Behaviour in Learning from Stochastic Examples by Ising Perceptrons

In a previous letter, we studied learning from stochastic examples by perceptrons with Ising weights in the framework of statistical mechanics. Under the one-step replica symmetry breaking ansatz, the behaviours of learning curves were classified according to some local property of the rules by which examples were drawn. Further, the conditions for the existence of the Perfect Learning together with other behaviors of the learning curves were given. In this paper, we give the detailed derivation about these results and further argument about the Perfect Learning together with extensive numerical calculations.

Spin glass states of the anti-Hopfield model

We study the Hopfield model with an opposite interactional sign by using a one-step replica symmetry breaking ansatz and the marginality condition. We show that this model belongs to spin glass models which have a dynamical phase transition, which is not associated with usual replica instability. Although this model is motivated by the observations about unlearning, it has various interesting aspects: correlations among interactions, a simple meaning as an optimization problem, and SK-like and non-SK-like spin glass phases depending on the number of `memorized' patterns.

Perfect learning and power law in learning from stochastic examples by Ising perceptrons: analysis under one-step replica symmetry breaking ansatz

We study learning from stochastic examples by perceptrons with Ising weights in the framework of statistical mechanics. Under the one-step replica symmetry breaking ansatz, the behaviour of learning curves are classified according to some local property of the rules by which examples are drawn. The conditions for the existence of perfect learning together with other behaviours of the learning curves are given precisely. The results agree with those obtained by Seung (1995) using a refined annealled approximation.