37 research outputs found
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
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
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
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
On the dynamics of the glass transition on Bethe lattices
The Glauber dynamics of disordered spin models with multi-spin interactions
on sparse random graphs (Bethe lattices) is investigated. Such models undergo a
dynamical glass transition upon decreasing the temperature or increasing the
degree of constrainedness. Our analysis is based upon a detailed study of large
scale rearrangements which control the slow dynamics of the system close to the
dynamical transition. Particular attention is devoted to the neighborhood of a
zero temperature tricritical point.
Both the approach and several key results are conjectured to be valid in a
considerably more general context.Comment: 56 pages, 38 eps figure
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
Exhaustive enumeration unveils clustering and freezing in random 3-SAT
We study geometrical properties of the complete set of solutions of the
random 3-satisfiability problem. We show that even for moderate system sizes
the number of clusters corresponds surprisingly well with the theoretic
asymptotic prediction. We locate the freezing transition in the space of
solutions which has been conjectured to be relevant in explaining the onset of
computational hardness in random constraint satisfaction problems.Comment: 4 pages, 3 figure
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
Geometrical organization of solutions to random linear Boolean equations
The random XORSAT problem deals with large random linear systems of Boolean
variables. The difficulty of such problems is controlled by the ratio of number
of equations to number of variables. It is known that in some range of values
of this parameter, the space of solutions breaks into many disconnected
clusters. Here we study precisely the corresponding geometrical organization.
In particular, the distribution of distances between these clusters is computed
by the cavity method. This allows to study the `x-satisfiability' threshold,
the critical density of equations where there exist two solutions at a given
distance.Comment: 20 page