38 research outputs found
A microscopic description of the aging dynamics: fluctuation-dissipation relations, effective temperature and heterogeneities
We consider the dynamics of a diluted mean-field spin glass model in the
aging regime. The model presents a particularly rich heterogeneous behavior. In
order to catch this behavior, we perform a **spin-by-spin analysis** for a
**given disorder realization**. The results compare well with the outcome of a
static calculation which uses the ``survey propagation'' algorithm of Mezard,
Parisi, and Zecchina [Sciencexpress 10.1126/science.1073287 (2002)]. We thus
confirm the connection between statics and dynamics at the level of single
degrees of freedom. Moreover, working with single-site quantities, we can
introduce a new response-vs-correlation plot, which clearly shows how
heterogeneous degrees of freedom undergo coherent structural rearrangements.
Finally we discuss the general scenario which emerges from our work and
(possibly) applies to more realistic glassy models. Interestingly enough, some
features of this scenario can be understood recurring to thermometric
considerations.Comment: 4 pages, 5 figures (7 eps files
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-ways 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 non paramagnetic 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.Comment: 19 pages, 13 figure
Glassy Critical Points and Random Field Ising Model
We consider the critical properties of points of continuous glass transition
as one can find in liquids in presence of constraints or in liquids in porous
media. Through a one loop analysis we show that the critical Replica Field
Theory describing these points can be mapped in the -Random Field Ising
Model. We confirm our analysis studying the finite size scaling of the -spin
model defined on sparse random graph, where a fraction of variables is frozen
such that the phase transition is of a continuous kind.Comment: The paper has been completely revised. A completely new part with
simulations of a p-spin glass model on random graph has been included. An
appendix with the Mathematica worksheet used in the calculation of the
diagrams has also been adde
Entropic long range order in a 3D spin glass model
We uncover a new kind of entropic long range order in finite dimensional spin
glasses. We study the link-diluted version of the Edwards-Anderson spin glass
model with bimodal couplings (J=+/-1) on a 3D lattice. By using exact reduction
algorithms, we prove that there exists a region of the phase diagram (at zero
temperature and link density low enough), where spins are long range
correlated, even if the ground states energy stiffness is null. In other words,
in this region twisting the boundary conditions cost no energy, but spins are
long range correlated by means of pure entropic effects.Comment: 15 pages, 6 figures. v3: added a phase diagram for ferromagnetically
biased coupling
Mosaic length and finite interaction-range effects in a one dimensional random energy model
In this paper we study finite interaction range corrections to the mosaic
picture of the glass transition as emerges from the study of the Kac limit of
large interaction range for disordered models. To this aim we consider point to
set correlation functions, or overlaps, in a one dimensional random energy
model as a function of the range of interaction. In the Kac limit, the mosaic
length defines a sharp first order transition separating a high overlap phase
from a low overlap one. Correspondingly we find that overlap curves as a
function of the window size and different finite interaction ranges cross
roughly at the mosaic lenght. Nonetheless we find very slow convergence to the
Kac limit and we discuss why this could be a problem for measuring the mosaic
lenght in realistic models.Comment: 18 pages, 7 figures, contribution for the special issue "Viewing the
World through Spin Glasses" in honour of Professor David Sherringto
Instability of one-step replica-symmetry-broken phase in satisfiability problems
We reconsider the one-step replica-symmetry-breaking (1RSB) solutions of two
random combinatorial problems: k-XORSAT and k-SAT. We present a general method
for establishing the stability of these solutions with respect to further steps
of replica-symmetry breaking. Our approach extends the ideas of [A.Montanari
and F. Ricci-Tersenghi, Eur.Phys.J. B 33, 339 (2003)] to more general
combinatorial problems.
It turns out that 1RSB is always unstable at sufficiently small clauses
density alpha or high energy. In particular, the recent 1RSB solution to 3-SAT
is unstable at zero energy for alpha< alpha_m, with alpha_m\approx 4.153. On
the other hand, the SAT-UNSAT phase transition seems to be correctly described
within 1RSB.Comment: 26 pages, 7 eps figure
Clusters of solutions and replica symmetry breaking in random k-satisfiability
We study the set of solutions of random k-satisfiability formulae through the
cavity method. It is known that, for an interval of the clause-to-variables
ratio, this decomposes into an exponential number of pure states (clusters). We
refine substantially this picture by: (i) determining the precise location of
the clustering transition; (ii) uncovering a second `condensation' phase
transition in the structure of the solution set for k larger or equal than 4.
These results both follow from computing the large deviation rate of the
internal entropy of pure states. From a technical point of view our main
contributions are a simplified version of the cavity formalism for special
values of the Parisi replica symmetry breaking parameter m (in particular for
m=1 via a correspondence with the tree reconstruction problem) and new large-k
expansions.Comment: 30 pages, 14 figures, typos corrected, discussion of appendix C
expanded with a new figur
On the cavity method for decimated random constraint satisfaction problems and the analysis of belief propagation guided decimation algorithms
We introduce a version of the cavity method for diluted mean-field spin
models that allows the computation of thermodynamic quantities similar to the
Franz-Parisi quenched potential in sparse random graph models. This method is
developed in the particular case of partially decimated random constraint
satisfaction problems. This allows to develop a theoretical understanding of a
class of algorithms for solving constraint satisfaction problems, in which
elementary degrees of freedom are sequentially assigned according to the
results of a message passing procedure (belief-propagation). We confront this
theoretical analysis to the results of extensive numerical simulations.Comment: 32 pages, 24 figure
Local overlaps, heterogeneities and the local fluctuation dissipation relations
In this paper I introduce the probability distribution of the local overlap
in spin glasses. The properties of the local overlaps are studied in details.
These quantities are related to the recently proposed local version of the
fluctuation dissipation relations: using the general principle of stochastic
stability these local fluctuation dissipation relations can be proved in a way
that is very similar to the usual proof of the fluctuation dissipation
relations for intensive quantities. The local overlap and its probability
distribution play a crucial role in this proof. Similar arguments can be used
to prove that all sites in an aging experiment stay at the same effective
temperature at the same time.Comment: 14 pages, no figure
Aging dynamics of heterogeneous spin models
We investigate numerically the dynamics of three different spin models in the
aging regime. Each of these models is meant to be representative of a distinct
class of aging behavior: coarsening systems, discontinuous spin glasses, and
continuous spin glasses. In order to study dynamic heterogeneities induced by
quenched disorder, we consider single-spin observables for a given disorder
realization. In some simple cases we are able to provide analytical predictions
for single-spin response and correlation functions.
The results strongly depend upon the model considered. It turns out that, by
comparing the slow evolution of a few different degrees of freedom, one can
distinguish between different dynamic classes. As a conclusion we present the
general properties which can be induced from our results, and discuss their
relation with thermometric arguments.Comment: 39 pages, 36 figure