arXiv subjects

Tommaso Rizzo

Publications and source records attributed to Tommaso Rizzo.

At least 55 records · Page 3Linked to original sources

arXiv2014

A replica trick for rare samples

In the context of disordered systems with quenched Hamiltonians I address the problem of characterizing rare samples where the thermal average of a specific observable has a value different from the typical one. These rare samples can be selected through a variation of the replica trick which amounts to replicate the system and divide the replicas in two groups containing respectively $M$ and $-M$ replicas. Replicas in the first (second) group experience an positive (negative) small field $O(1/M)$ conjugate to the observable considered and the $M \rightarrow \infty$ limit is to be taken in the end. Applications to the random-field Ising model and to the Sherrington-Kirkpatrick model are discussed.

cond-mat.dis-nn↗

arXiv2013

Temperature chaos and quenched heterogeneities

We present a treatable generalization of the Sherrington Kirkpatrick model which introduces correlations in the elements of the coupling matrix through multiplicative disorder on the single variables and investigate the consequences on the phase diagram. We define a generalized $q_{EA}$ parameter and test the structural stability of the SK results in this correlated case evaluating the AT-line of the model. As main result we demonstrate the increase of temperature chaos effects due to heterogeneities.

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arXiv2013

Finite size corrections to disordered systems on Erdös-Rényi random graphs

We study the finite size corrections to the free energy density in disorder spin systems on sparse random graphs, using both replica theory and cavity method. We derive an analytical expressions for the $O(1/N)$ corrections in the replica symmetric phase as a linear combination of the free energies of open and closed chains. We perform a numerical check of the formulae on the Random Field Ising Model at zero temperature, by computing finite size corrections to the ground state energy density.

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arXiv2013

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 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, it 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 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 1RBS 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.

cond-mat.dis-nn↗

arXiv2013

Towards a Theory of the Glass Crossover

The standard field-theoretical procedure to study the effect of long wavelength fluctuations on a genuine second-order phase transition is applied to the Mode-Coupling-Theory (MCT) dynamical singularity at $T_c$ in the $β$ regime. Technically this is achieved by a dynamical field-theoretical decoration of MCT that can be studied by a loop expansion. An explicit computation shows that at all orders the leading contributions are the same of a dynamical stochastic glassy equation, {\it i.e} an extension of the standard MCT equation for the critical correlator with local random fluctuations of the separation parameter. It is suggested that the equation is an essential ingredient in the process that turns the singularity at $T_c$ into a dynamical crossover to activated dynamics.

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arXiv2013

Super-Cooled Liquids: Equivalence between Mode-Coupling Theory and Replica Approach

We show that the replica approach to glassy dynamics provides, in spite of its static nature, a characterization of critical dynamics in the $β$-regime of super-cooled liquids that is equivalent to the one of Mode-Coupling-Theory, both qualitatively and quantitatively. The nature and extent of this equivalence is discussed in connection to the main open problems of the current theory.

cond-mat.dis-nn↗

arXiv2013

Critical Off-Equilibrium Dynamics in Glassy Systems

We consider off-equilibrium dynamics at the critical temperature in a class of glassy system. The off-equilibrium correlation and response functions obey a precise scaling form in the aging regime. The structure of the {\it equilibrium} replicated Gibbs free energy fixes the corresponding {\it off-equilibrium} scaling functions implicitly through two functional equations. The details of the model enter these equations only through the ratio $w_2/w_1$ of the cubic coefficients (proper vertexes) of the replicated Gibbs free energy. Therefore the off-equilibrium dynamical exponents are controlled by the very same parameter exponent $λ=w_2/w_1$ that determines equilibrium dynamics. We find approximate solutions to the equations and validate the theory by means of analytical computations and numerical simulations.

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arXiv2012

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 non-universal 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.

cond-mat.dis-nn↗

arXiv2012

A note on weakly discontinuous dynamical transitions

We analyze Mode Coupling discontinuous transition in the limit of vanishing discontinuity, approaching the so called "$A_3$" point. In these conditions structural relaxation and fluctuations appear to have universal form independent from the details of the system. The analysis of this limiting case suggests new ways for looking at the Mode Coupling equations in the general case.

cond-mat.stat-mech↗

arXiv2012

Replica Cluster Variational Method: the Replica Symmetric solution for the 2D random bond Ising model

We present and solve the Replica Symmetric equations in the context of the Replica Cluster Variational Method for the 2D random bond Ising model (including the 2D Edwards-Anderson spin glass model). First we solve a linearized version of these equations to obtain the phase diagrams of the model on the square and triangular lattices. In both cases the spin-glass transition temperatures and the tricritical point estimations improve largely over the Bethe predictions. Moreover, we show that this phase diagram is consistent with the behavior of inference algorithms on single instances of the problem. Finally, we present a method to consistently find approximate solutions to the equations in the glassy phase. The method is applied to the triangular lattice down to T=0, also in the presence of an external field.

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arXiv2012

Dynamical critical exponents for the mean-field Potts glass

In this paper we study the critical behaviour of the fully-connected p-colours Potts model at the dynamical transition. In the framework of Mode Coupling Theory (MCT), the time autocorrelation function displays a two step relaxation, with two exponents governing the approach to the plateau and the exit from it. Exploiting a relation between statics and equilibrium dynamics which has been recently introduced, we are able to compute the critical slowing down exponents at the dynamical transition with arbitrary precision and for any value of the number of colours p. When available, we compare our exact results with numerical simulations. In addition, we present a detailed study of the dynamical transition in the large p limit, showing that the system is not equivalent to a random energy model.

cond-mat.dis-nn↗

arXiv2012

Critical slowing down exponents in quenched disordered spin models for structural glasses: Random Orthogonal and related models

An important prediction of Mode-Coupling-Theory (MCT) is the relationship between the power- law decay exponents in the β regime. In the original structural glass context this relationship follows from the MCT equations that are obtained making rather uncontrolled approximations and λ has to be treated like a tunable parameter. It is known that a certain class of mean-field spin-glass models is exactly described by MCT equations. In this context, the physical meaning of the so called parameter exponent λ has recently been unveiled, giving a method to compute it exactly in a static framework. In this paper we exploit this new technique to compute the critical slowing down exponents in a class of mean-field Ising spin-glass models including, as special cases, the Sherrington-Kirkpatrick model, the p-spin model and the Random Orthogonal model.

cond-mat.dis-nn↗

arXiv2012

On the two-steps relaxation of mean-field glasses: p-spin model

Critical slowing down dynamics of supercooled glass-forming liquids is usually understood at the mean-field level in the framework of Mode Coupling Theory, providing a two-time relaxation scenario and power-law behaviors of the time correlation function at dynamic criticality. In this work we derive critical slowing down exponents of spin-glass models undergoing discontinuous transitions by computing their Gibbs free energy and connecting the dynamic behavior to static "in-state" properties. Both the spherical and Ising versions are considered and, in the simpler spherical case, a generalization to arbitrary schematic Mode Coupling kernels is presented. Comparison with dynamic results available in literature is performed. Analytical predictions for the Ising case are provided for any $p$.

cond-mat.dis-nn↗

arXiv2011

Field Theory of Fluctuations in Glasses

We develop a field-theoretical description of dynamical heterogeneities and fluctuations in supercooled liquids close to the (avoided) MCT singularity. Using quasi-equilibrium arguments we eliminate time from the description and we completely characterize fluctuations in the beta regime. We identify different sources of fluctuations and show that the most relevant ones are associated to variations of "self-induced disorder" in the initial condition of the dynamics. It follows that heterogeneites can be describes through a cubic field theory with an effective random field term. The phenomenon of perturbative dimensional reduction ensues, well known in random field problems, which implies an upper critical dimension of the theory equal to 8. We apply our theory to finite size scaling for mean-field systems and we test its prediction against numerical simulations.

cond-mat.stat-mech↗

arXiv2011

Replica Field Theory of the Dynamical Transition in Glassy Systems

The critical behaviour of the dynamical transition of glassy system is controlled by a Replica Symmetric action with n=1 replicas. The most divergent diagrams in the loop expansion correspond at all orders to the solutions of a stochastic equation leading to perturbative dimensional reduction. The theory describe accurately numerical simulations of mean-field models.

cond-mat.stat-mech↗

arXiv2011

A very fast inference algorithm for finite-dimensional spin glasses: Belief Propagation on the dual lattice

Starting from a Cluster Variational Method, and inspired by the correctness of the paramagnetic Ansatz (at high temperatures in general, and at any temperature in the 2D Edwards-Anderson model) we propose a novel message passing algorithm --- the Dual algorithm --- to estimate the marginal probabilities of spin glasses on finite dimensional lattices. We show that in a wide range of temperatures our algorithm compares very well with Monte Carlo simulations, with the Double Loop algorithm and with exact calculation of the ground state of 2D systems with bimodal and Gaussian interactions. Moreover it is usually 100 times faster than other provably convergent methods, as the Double Loop algorithm.

cond-mat.dis-nn↗

arXiv2010

Critical behaviour of large scale dynamical heterogeneities in glasses: a complete theory

In this talk I will present a complete theory for the behaviour of large-scale dynamical heterogeneities in glasses. Following the work arXiv:1001.1746 I will show that we can write a (physically motivated) simple stochastic differential equation that is potentially able to explain the behaviour of large scale dynamical heterogeneities in glasses. It turns out that this behaviour is in the same universality class of the dynamics near the endpoint of a metastable phase in a disordered system, as far as reparametrization invariant quantities are concerned. Therefore Large scale dynamical heterogeneities in glasses have many points in contact with the Barkhausen noise. Numerical verifications of this theory have not yet done, but they are quite possible.

cond-mat.dis-nn↗

arXiv2010

Chaos in Temperature in Diluted Mean-Field Spin-Glasses

We consider the problem of temperature chaos in mean-field spin-glass models defined on random lattices with finite connectivity. By means of an expansion in the order parameter we show that these models display a much stronger chaos effect than the fully connected Sherrington-Kirkpatrick model with the exception of the Bethe lattice with bimodal distribution of the couplings.

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