Tommaso Rizzo

Critical slowing down exponents of mode coupling theory

A method is provided to compute the exponent parameter λ yielding the dynamic exponents of critical slowing down in mode coupling theory. It is independent from the dynamic approach and based on the formulation of an effective static field theory. Expressions of λ in terms of third order coefficients of the action expansion or, equivalently, in terms of six point cumulants are provided. Applications are reported to a number of mean-field models: with hard and soft variables and both fully connected and dilute interactions. Comparisons with existing results for the Potts glass model, the random orthogonal model, hard and soft-spin Sherrington-Kirkpatrick, and p-spin models are presented.

Critical dynamics in glassy systems.

Critical dynamics in various glass models, including those described by mode-coupling theory, is described by scale-invariant dynamical equations with a single nonuniversal quantity, i.e., the so-called parameter exponent that determines all the dynamical critical exponents. We show that these equations follow from the structure of the static replicated Gibbs free energy near the critical point. In particular, the exponent parameter is given by the ratio between two cubic proper vertexes that can be expressed as six-point cumulants measured in a purely static framework.

Large deviations in the free energy of mean-field spin glasses

We compute analytically the probability distribution of large deviations in the spin-glass free energy for the Sherrington-Kirkpatrick mean-field model; i.e., we compute the exponentially small probability of finding a system with intensive free energy smaller than the most likely one. This result is obtained by computing the average value of the partition function to the power n as a function of n. At zero temperature this absolute prediction displays a remarkable quantitative agreement with the numerical data.

Spin-glass complexity

We study the quenched complexity in spin-glass mean-field models satisfying the Becchi-Rouet-Stora-Tyutin supersymmetry. The outcome of such study, consistent with recent numerical results, allows, in principle, to conjecture the absence of any supersymmetric contribution to the complexity in the Sherrington-Kirkpatrick model. The same analysis can be applied to any model with a full replica symmetry breaking phase, e.g., the Ising p-spin model below the Gardner temperature. The existence of different solutions, breaking the supersymmetry, is also discussed.

The complexity of the spherical p-spin spin glass model, revisited

Abstract.Some questions concerning the calculation of the number of “physical” (metastable) states or complexity of the spherical p-spin spin glass model are reviewed and examined further. Particular attention is focused on the general calculation procedure which is discussed step-by-step.

Characterizing and improving generalized belief propagation algorithms on the 2D Edwards-Anderson model

We study the performance of different message passing algorithms in the two-dimensional Edwards–Anderson model. We show that the standard belief propagation (BP) algorithm converges only at high temperature to a paramagnetic solution. Then, we test a generalized belief propagation (GBP) algorithm, derived from a cluster variational method (CVM) at the plaquette level. We compare its performance with BP and with other algorithms derived under the same approximation: double loop (DL) and a two-way message passing algorithm (HAK). The plaquette-CVM approximation improves BP in at least three ways: the quality of the paramagnetic solution at high temperatures, a better estimate (lower) for the critical temperature, and the fact that the GBP message passing algorithm converges also to nonparamagnetic solutions. The lack of convergence of the standard GBP message passing algorithm at low temperatures seems to be related to the implementation details and not to the appearance of long range order. In fact, we prove that a gauge invariance of the constrained CVM free energy can be exploited to derive a new message passing algorithm which converges at even lower temperatures. In all its region of convergence this new algorithm is faster than HAK and DL by some orders of magnitude.

Replica-symmetry-breaking transitions and off-equilibrium dynamics.

I consider branches of replica-symmetry-breaking (RSB) solutions in glassy systems that display a dynamical transition at a temperature T_{d} characterized by a mode-coupling-theory dynamical behavior. Below T_{d} these branches of solutions are considered to be relevant to the system complexity and to off-equilibrium dynamics. Under general assumptions I argue that near T_{d} it is not possible to stabilize the one-step (1RSB) solution beyond the marginal point by making a full RSB (FRSB) ansatz. However, depending on the model, there may exist a temperature T strictly lower than T_{d} below which the 1RSB branch can be continued to a FRSB branch. Such a temperature certainly exists for models that display the so-called Gardner transition and in this case T_{G}<T_<T_{d}. An analytical study in the context of the truncated model reveals that the FRSB branch of solutions below T is characterized by a two-plateau structure and it ends where the first plateau disappears. These general features are confirmed in the context of the Ising p-spin model with p=3 by means of a numerical solution of the FRSB equations. The results are discussed in connection with off-equilibrium dynamics within Cugliandolo-Kurchan theory. In this context I assume that the RSB solution relevant for off-equilibrium dynamics is the 1RSB marginal solution in the whole range (T ,T_{d}) and it is the end point of the FRSB branch for T<T. Remarkably, under these assumptions it can be argued that T marks a qualitative change in off-equilibrium dynamics in the sense that the decay of various dynamical quantities changes from power law to logarithmic.

Large deviations of the free energy in diluted mean-field spin-glass

Sample-to-sample free-energy fluctuations in spin-glasses display a markedly different behaviour in finite-dimensional and fully connected models, namely Gaussian versus non-Gaussian. Spin-glass models defined on various types of random graphs are in an intermediate situation between these two classes of models and we investigate whether the nature of their free-energy fluctuations is Gaussian or not. It has been argued that Gaussian behaviour is present whenever the interactions are locally non-homogeneous, i.e. in most cases with the notable exception of models with fixed connectivity and random couplings . We confirm these expectations by means of various analytical results concerning the large deviations of the free energy. In particular we unveil the connection between the spatial fluctuations of the populations of fields defined at different sites of the lattice and the Gaussian nature of the free-energy fluctuations. In contrast, on locally homogeneous lattices the populations do not fluctuate over the sites and as a consequence the small deviations of the free energy are non-Gaussian and scale as in the Sherrington–Kirkpatrick model.

Quenched computation of the dependence of complexity on the free energy in the Sherrington-Kirkpatrick model

The quenched computation of the complexity in the Sherrington-Kirkpatrick model is presented. A modified Full Replica Symmetry Breaking Ansatz is introduced in order to study the complexity dependence on the free energy. Such an Ansatz corresponds to require Becchi-Rouet-Stora-Tyutin supersymmetry. The complexity computed this way is the Legendre transform of the free energy averaged over the quenched disorder. The stability analysis shows that this complexity is inconsistent at any free energy level but the equilibirum one. The further problem of building a physically well defined solution not invariant under supersymmetry and predicting an extensive number of metastable states is also discussed.

Replica Cluster Variational Method

We present a general formalism to make the Replica-Symmetric and Replica-Symmetry-Breaking ansatz in the context of Kikuchi’s Cluster Variational Method (CVM). Using replicas and the message-passing formulation of CVM we obtain a variational expression of the replicated free energy of a system with quenched disorder, both averaged and on a single sample, and make the hierarchical ansatz using functionals of functions of fields to represent the messages. We obtain a set of integral equations for the message functionals. The main difference with the Bethe case is that the functionals appear in the equations in implicit form and are not positive definite, thus standard iterative population dynamic algorithms cannot be used to determine them. In the simplest cases the solution could be obtained iteratively using Fourier transforms.We begin to study the method considering the plaquette approximation to the averaged free energy of the Edwards-Anderson model in the paramagnetic Replica-Symmetric phase. In two dimensions we find that the spurious spin-glass phase transition of the Bethe approximation disappears and the paramagnetic phase is stable down to zero temperature on the square lattice for different random interactions. The quantitative estimates of the free energy and of various other quantities improve those of the Bethe approximation. The plaquette approximation fails to predict a second-order spin-glass phase transition on the cubic 3D lattice but yields good results in dimension four and higher. We provide the physical interpretation of the beliefs in the replica-symmetric phase as disorder distributions of the local Hamiltonian. The messages instead do not admit such an interpretation and indeed they cannot be represented as populations in the spin-glass phase at variance with the Bethe approximation.The approach can be used in principle to study the phase diagram of a wide range of disordered systems and it is also possible that it can be used to get quantitative predictions on single samples. These further developments present however great technical challenges.