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ABSTRACT: 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
05/2013;
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ABSTRACT: 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.
The Journal of chemical physics 02/2013; 138(6):064504. · 3.09 Impact Factor
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ABSTRACT: 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.
Physical Review E 01/2013; 87(1-1):012101. · 2.26 Impact Factor
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ABSTRACT: 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 $\lambda=w_2/w_1$ that determines equilibrium
dynamics. We find approximate solutions to the equations and validate the
theory by means of analytical computations and numerical simulations.
12/2012;
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ABSTRACT: 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.
KeywordsSpin glasses-Cluster variation method-Replica method
Journal of Statistical Physics 04/2012; 139(3):375-416. · 1.40 Impact Factor
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ABSTRACT: 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.
04/2012;
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ABSTRACT: An important prediction of Mode-Coupling-Theory (MCT) is the relationship
between the power- law decay exponents in the {\beta} regime. In the original
structural glass context this relationship follows from the MCT equations that
are obtained making rather uncontrolled approximations and {\lambda} 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 {\lambda} 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.
03/2012;
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ABSTRACT: 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.
Phys. Rev. E. 02/2012; 85(5).
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ABSTRACT: 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 $p$.
02/2012;
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ABSTRACT: 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 two-dimensional (2D) Edwards-Anderson (EA) model] we propose a message-passing algorithm--the dual algorithm--to estimate the marginal probabilities of spin glasses on finite-dimensional lattices. We use the EA models in 2D and 3D as benchmarks. The dual algorithm improves the Bethe approximation, and we show that in a wide range of temperatures (compared to the Bethe critical temperature) 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. In 2D and 3D the quality of the inference deteriorates only where the correlation length becomes very large, i.e., at low temperatures in 2D and close to the critical temperature in 3D.
Physical Review E 10/2011; 84(4 Pt 2):046706. · 2.26 Impact Factor
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ABSTRACT: 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.
05/2011;
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ABSTRACT: 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.
02/2011;
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CoRR. 01/2011; abs/1110.1259.
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ABSTRACT: 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. Comment: Talk given by Giorgio Parisi at StatphysHK, Hong-Kong, July 2010, 14 pages, 6 figures
08/2010;
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ABSTRACT: 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 a bimodal distribution of the couplings.
Journal of Physics A Mathematical and Theoretical 05/2010; 43(23):235003. · 1.56 Impact Factor
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ABSTRACT: In this letter we show how to perform a systematic perturbative approach for the mode-coupling theory. The results coincide with those obtained via the replica approach. The upper critical dimension turns out to be always 8 and the correlations have a double pole in momentum space in perturbations theory. Non-perturbative effects are found to be very important. We suggest a possible framework to compute these effects. Comment: 5 pages, 1 figure
01/2010;
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ABSTRACT: Sample-to-sample free energy fluctuations in spin-glasses display a markedly
different behaviour in finite-dimensional and fully-connected models, namely
Gaussian vs. 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 $J_{ij}=\pm
\tilde{J}$. We confirm these expectation by means of various analytical
results. In particular we unveil the connection between the spatial
fluctuations of the populations of populations of fields defined at different
sites of the lattice and the Gaussian nature of the free-energy fluctuations.
On the contrary 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 scales as in the Sherrington-Kirkpatrick model.
10/2009;
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ABSTRACT: We compute the probability of positive large deviations of the free energy
per spin in mean-field Spin-Glass models. The probability vanishes in the
thermodynamic limit as $P(\Delta f) \propto \exp[-N^2 L_2(\Delta f)]$. For the
Sherrington-Kirkpatrick model we find $L_2(\Delta f)=O(\Delta f)^{12/5}$ in
good agreement with numerical data and with the assumption that typical small
deviations of the free energy scale as $N^{1/6}$. For the spherical model we
find $L_2(\Delta f)=O(\Delta f)^{3}$ in agreement with recent findings on the
fluctuations of the largest eigenvalue of random Gaussian matrices. The
computation is based on a loop expansion in replica space and the non-gaussian
behaviour follows in both cases from the fact that the expansion is divergent
at all orders. The factors of the leading order terms are obtained resumming
appropriately the loop expansion and display universality, pointing to the
existence of a single universal distribution describing the small deviations of
any model in the full-Replica-Symmetry-Breaking class.
01/2009;
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ABSTRACT: We consider the probability distribution of large deviations in the
spin-glass free energy for the Sherrington-Kirkpatrick mean field model, i.e.
the exponentially small probability of finding a system with intensive free
energy smaller than the most likely one. This result is obtained by computing
$\Phi(n,T)=T \bar{Z^n}/ n$, i.e. the average value of the partition function to
the power $n$ as a function of $n$. We study in full details the phase diagram
of $\Phi(n,T)$ in the $(n,T)$ plane computing in particular the stability of
the replica-symmetric solution. At low temperatures we compute $\Phi(n,T)$ in
series of $n$ and $\tau=T_c-T$ at high orders using the standard hierarchical
ansatz and confirm earlier findings on the $O(n^5)$ scaling. We prove that the
$O(n^5)$ scaling is valid at all orders and obtain an exact expression for the
coefficient in term of the function $q(x)$. Resumming the series we obtain the
large deviations probability at all temperatures. At zero temperature the
analytical prediction displays a remarkable quantitative agreement with the
numerical data. A similar computation for the simpler spherical model is also
performed and the connection between large and small deviations is discussed.
11/2008;
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ABSTRACT: 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.
Physical Review Letters 10/2008; 101(11):117205. · 7.37 Impact Factor
