144 research outputs found
Dynamics of dilute disordered models: a solvable case
We study the dynamics of a dilute spherical model with two body interactions
and random exchanges. We analyze the Langevin equations and we introduce a
functional variational method to study generic dilute disordered models. A
crossover temperature replaces the dynamic transition of the fully-connected
limit. There are two asymptotic regimes, one determined by the central band of
the spectral density of the interactions and a slower one determined by
localized configurations on sites with high connectivity. We confront the
behavior of this model to the one of real glasses.Comment: 7 pages, 4 figures. Clarified, final versio
Systematic perturbation approach for a dynamical scaling law in a kinetically constrained spin model
The dynamical behaviours of a kinetically constrained spin model
(Fredrickson-Andersen model) on a Bethe lattice are investigated by a
perturbation analysis that provides exact final states above the nonergodic
transition point. It is observed that the time-dependent solutions of the
derived dynamical systems obtained by the perturbation analysis become
systematically closer to the results obtained by Monte Carlo simulations as the
order of a perturbation series is increased. This systematic perturbation
analysis also clarifies the existence of a dynamical scaling law, which
provides a implication for a universal relation between a size scale and a time
scale near the nonergodic transition.Comment: 17 pages, 7 figures, v2; results have been refined, v3; A figure has
been modified, v4; results have been more refine
From Large Scale Rearrangements to Mode Coupling Phenomenology
We consider the equilibrium dynamics of Ising spin models with multi-spin
interactions on sparse random graphs (Bethe lattices). Such models undergo a
mean field glass transition upon increasing the graph connectivity or lowering
the temperature. Focusing on the low temperature limit, we identify the large
scale rearrangements responsible for the dynamical slowing-down near the
transition. We are able to characterize exactly the dynamics near criticality
by analyzing the statistical properties of such rearrangements. Our approach
can be generalized to a large variety of glassy models on sparse random graphs,
ranging from satisfiability to kinetically constrained models.Comment: 4 pages, 4 figures, minor corrections, accepted versio
Sparse random matrices: the eigenvalue spectrum revisited
We revisit the derivation of the density of states of sparse random matrices.
We derive a recursion relation that allows one to compute the spectrum of the
matrix of incidence for finite trees that determines completely the low
concentration limit. Using the iterative scheme introduced by Biroli and
Monasson [J. Phys. A 32, L255 (1999)] we find an approximate expression for the
density of states expected to hold exactly in the opposite limit of large but
finite concentration. The combination of the two methods yields a very simple
simple geometric interpretation of the tails of the spectrum. We test the
analytic results with numerical simulations and we suggest an indirect
numerical method to explore the tails of the spectrum.Comment: 18 pages, 7 figures. Accepted version, minor corrections, references
adde
Analytic determination of dynamical and mosaic length scales in a Kac glass model
We consider a disordered spin model with multi-spin interactions undergoing a
glass transition. We introduce a dynamic and a static length scales and compute
them in the Kac limit (long--but--finite range interactions). They diverge at
the dynamic and static phase transition with exponents (respectively) -1/4 and
-1. The two length scales are approximately equal well above the mode coupling
transition. Their discrepancy increases rapidly as this transition is
approached. We argue that this signals a crossover from mode coupling to
activated dynamics.Comment: 4 pages, 4 eps figures. New version conform to the published on
Approximation schemes for the dynamics of diluted spin models: the Ising ferromagnet on a Bethe lattice
We discuss analytical approximation schemes for the dynamics of diluted spin
models. The original dynamics of the complete set of degrees of freedom is
replaced by a hierarchy of equations including an increasing number of global
observables, which can be closed approximately at different levels of the
hierarchy. We illustrate this method on the simple example of the Ising
ferromagnet on a Bethe lattice, investigating the first three possible
closures, which are all exact in the long time limit, and which yield more and
more accurate predictions for the finite-time behavior. We also investigate the
critical region around the phase transition, and the behavior of two-time
correlation functions. We finally underline the close relationship between this
approach and the dynamical replica theory under the assumption of replica
symmetry.Comment: 21 pages, 5 figure
Relationship between clustering and algorithmic phase transitions in the random k-XORSAT model and its NP-complete extensions
We study the performances of stochastic heuristic search algorithms on
Uniquely Extendible Constraint Satisfaction Problems with random inputs. We
show that, for any heuristic preserving the Poissonian nature of the underlying
instance, the (heuristic-dependent) largest ratio of constraints per
variables for which a search algorithm is likely to find solutions is smaller
than the critical ratio above which solutions are clustered and
highly correlated. In addition we show that the clustering ratio can be reached
when the number k of variables per constraints goes to infinity by the
so-called Generalized Unit Clause heuristic.Comment: 15 pages, 4 figures, Proceedings of the International Workshop on
Statistical-Mechanical Informatics, September 16-19, 2007, Kyoto, Japan; some
imprecisions in the previous version have been correcte
Clustering of solutions in the random satisfiability problem
Using elementary rigorous methods we prove the existence of a clustered phase
in the random -SAT problem, for . In this phase the solutions are
grouped into clusters which are far away from each other. The results are in
agreement with previous predictions of the cavity method and give a rigorous
confirmation to one of its main building blocks. It can be generalized to other
systems of both physical and computational interest.Comment: 4 pages, 1 figur
Message passing for vertex covers
Constructing a minimal vertex cover of a graph can be seen as a prototype for
a combinatorial optimization problem under hard constraints. In this paper, we
develop and analyze message passing techniques, namely warning and survey
propagation, which serve as efficient heuristic algorithms for solving these
computational hard problems. We show also, how previously obtained results on
the typical-case behavior of vertex covers of random graphs can be recovered
starting from the message passing equations, and how they can be extended.Comment: 25 pages, 9 figures - version accepted for publication in PR
Surface Tension in Kac Glass Models
In this paper we study a distance-dependent surface tension, defined as the
free-energy cost to put metastable states at a given distance. This will be
done in the framework of a disordered microscopic model with Kac interactions
that can be solved in the mean-field limit.Comment: 13 pages, 6 figure
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