arXiv subjects
Publications and source records attributed to Giorgio Parisi.
In this paper we study the critical properties of a finite dimensional generalization of the p-spin model. We find evidence that in dimension three, contrary to its mean field limit, the glass transition is associated to a diverging susceptibility (and correlation length).
In this work we study numerically a short range p-spin glass model in three dimensions. The behaviour of the model appears to be remarkably different from mean field predictions. In fact it shares some features typical of models with full replica-symmetry breaking (FRSB). Nevertheless, we believe that the transition that we study is intrinsically different from the FRSB and basically due to non-perturbative contributions. We study both the statics and the dynamics of the system which seem to confirm our conjectures.
A simple model to investigate the long time dynamics of glass-formers is presented and applied to study a Lennard-Jones system in supercooled and glassy phases. According to our model, the point representing the system in the configurational phase space performs harmonic vibrations around (and activated jumps between) minima pertaining to a connected network. Exploiting the model, in agreement with the experimental results, we find evidence for: i) stretched relaxational dynamics; ii) a strong T-dependence of the stretching parameter; iii) breakdown of the Stokes-Einstein law.
Attractors in asymmetric neural networks with deterministic parallel dynamics were shown to present a "chaotic" regime at symmetry eta < 0.5, where the average length of the cycles increases exponentially with system size, and an oscillatory regime at high symmetry, where the typical length of the cycles is 2. We show, both with analytic arguments and numerically, that there is a sharp transition, at a critical symmetry $\e_c=0.33$, between a phase where the typical cycles have length 2 and basins of attraction of vanishing weight and a phase where the typical cycles are exponentially long with system size, and the weights of their attraction basins are distributed as in a Random Map with reversal symmetry. The time-scale after which cycles are reached grows exponentially with system size $N$, and the exponent vanishes in the symmetric limit, where $T\propto N^{2/3}$. The transition can be related to the dynamics of the infinite system (where cycles are never reached), using the closing probabilities as a tool. We also study the relaxation of the function $E(t)=-1/N\sum_i |h_i(t)|$, where $h_i$ is the local field experienced by the neuron $i$. In the symmetric system, it plays the role of a Ljapunov function which drives the system towards its minima through steepest descent. This interpretation survives, even if only on the average, also for small asymmetry. This acts like an effective temperature: the larger is the asymmetry, the faster is the relaxation of $E$, and the higher is the asymptotic value reached. $E$ reachs very deep minima in the fixed points of the dynamics, which are reached with vanishing probability, and attains a larger value on the typical attractors, which are cycles of length 2.
In this note we formulate a finite dimensional generalization of the Random Energy Model (REM) where we introduce a geometry and spatial correlations between energies. We study the model in dimension one by transfer matrix techniques and we look at the crossover from one dimensional to mean-field behaviour. In a first version of the model the mean field limit reproduces the behaviour of the original REM, while a second version of the model exhibits a first order phase transition with a finite latent heat.
In this note we put forward a conjecture on the average optimal length for bipartite matching with a finite number of elements where the different lengths are independent one from the others and have an exponential distribution.
By means of an effective potential associated to a constrained equilibrium measure and apt to study frozen systems, we investigate glassy freezing in simple liquids in the hypernetted chain (HNC) approximation. Differently from other classical approximations of liquid theory, freezing is naturally embedded in the HNC approximation. We get a detailed description of the freezing transition that is analogous to the one got in a large class of mean-field long range spin glass. We compare our findings with Monte Carlo simulations of the same system and conclude that many of the qualitative features of the transition are captured by the approximated theory.
In this paper we review the predictions of the replica approach on the probability distribution of the overlaps among replicas and on the sample to sample fluctuations of this probability. We stress the role of replica equivalence in obtaining relations which do not depend on the form of replica symmetry breaking. A comparison is done with the results obtained with a different rigorous approach. The role of the ultrametricity and of other algebraic properties in discussed. It is shown that the ultrametric solution can be obtained from a variational principle.
In off-equilibrium dynamics we define a dynamical correlation length which is proportional to the size of the region in which the atoms move in a correlated way. General arguments indicate that this dynamical correlation length diverges at large times in the glassy phase. Numerical simulations for binary mixtures point toward the correctness of this prediction.
We review the predictions of the replica approach both for the statics and for the off-equilibrium dynamics. We stress the importance of the Cugliandolo-Kurchan off-equilibrium fluctuation-dissipation relation in providing a bridge between the statics and the dynamics. We present numerical evidence for the correctness of these relations. This approach allows an experimental determination of the basic parameters of the replica theory.
In this note we apply some theoretical predictions that arise in the mean field framework for a large class of infinite range models to structural glasses and we present a first comparison of these predictions with numerical results.
We present a variational approach for directed polymers in $D$ transversal dimensions which is used to compute the corrections to the mean field theory predictions with broken replica symmetry. The trial function is taken to be a symmetrized version of the mean-field solution, which is known to be exact for $D=\infty$. We compute the free energy corresponding to that function and show that the finite-$D$ corrections behave like $D^{-4/3}$. It means that the expansion in powers of 1/D should be used with great care here. We hope that the techniques developed in this note will be useful also in the study of spin glasses.
We study the glassy transition for simple liquids in the hypernetted chain (HNC) approximation by means of an effective potential recently introduced. Integrating the HNC equations for hard spheres, we find a transition scenario analogous to that of the long range disordered systems with ``one step replica symmetry breaking''. Our result agree qualitatively with Monte Carlo simulations of three dimensional hard spheres.
We study the dynamical behavior of the Sherrington Kirkpatrick model. Thanks to the APE supercomputer we are able to analyze large lattice volumes, and to investigate the low $T$ region. We present a determination of the remnant magnetization and of its time decay exponent, of the energy time decay exponent, and we discuss aging phenomena in the model.
In this talk I will review some of the recent applications of the replica theory to glasses. I will firstly describe the basic assumptions and I will show that they can be considered as a precise reformulations of old ideas. The relation of this approach with the mode-coupling theory will be shortly discussed. I will present numerical simulations for binary mixtures. The results of these simulations point toward the correctness of the replica approach to glasses. I will describe the results of off-equilibrium simulations for large systems, in which the aging dynamics is studied.
We study the phase diagram of glassy systems in presence of an attractive coupling among real replicas. We find competition among a localized and a delocalized phase, that are separated by a coexistence line as in ordinary first order phase transitions. The coexistence line terminates in a critical point. We present numerical simulations for binary glasses in which show that this scenario is realized.
We study the four dimensional Gaussian spin glass in presence of a magnetic field. Using off-equilibrium numerical simulations we have found that the probability distribution of the overlaps is built in the same way as that of the Mean Field approximation with replica symmetry breaking. Finally we have studied the violation of the fluctuation-dissipation theorem in presence of magnetic field.
We analyse the statistical physics of a two dimensional lattice based gas with long range interactions. The particles interact in a way analogous to Queens on a chess board. The long range nature of the interaction gives the mathematics of the problem a simple geometric structure which simplifies both the analytic and numerical study of the system. We present some analytic calculations for the statics of the problem and also we perform Monte Carlo simulations which exhibit a dynamical transition between a high temperature liquid regime and a low temperature glassy regime exhibiting aging in the two time correlation functions.