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

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

Patch-repetition correlation length in glassy systems

We obtain the patch-repetition entropy Sigma within the Random First Order Transition theory (RFOT) and for the square plaquette system, a model related to the dynamical facilitation theory of glassy dynamics. We find that in both cases the entropy of patches of linear size l, Sigma(l), scales as s_c l^d+A l^{d-1} down to length-scales of the order of one, where A is a positive constant, s_c is the configurational entropy density and d the spatial dimension. In consequence, the only meaningful length that can be defined from patch-repetition is the cross-over length xi=A/s_c. We relate xi to the typical length-scales already discussed in the literature and show that it is always of the order of the largest static length. Our results provide new insights, which are particularly relevant for RFOT theory, on the possible real space structure of super-cooled liquids. They suggest that this structure differs from a mosaic of different patches having roughly the same size.Comment: 6 page

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

Relaxation and Metastability in the RandomWalkSAT search procedure

An analysis of the average properties of a local search resolution procedure for the satisfaction of random Boolean constraints is presented. Depending on the ratio alpha of constraints per variable, resolution takes a time T_res growing linearly (T_res \sim tau(alpha) N, alpha < alpha_d) or exponentially (T_res \sim exp(N zeta(alpha)), alpha > alpha_d) with the size N of the instance. The relaxation time tau(alpha) in the linear phase is calculated through a systematic expansion scheme based on a quantum formulation of the evolution operator. For alpha > alpha_d, the system is trapped in some metastable state, and resolution occurs from escape from this state through crossing of a large barrier. An annealed calculation of the height zeta(alpha) of this barrier is proposed. The polynomial/exponentiel cross-over alpha_d is not related to the onset of clustering among solutions.Comment: 23 pages, 11 figures. A mistake in sec. IV.B has been correcte

Rigorous Inequalities between Length and Time Scales in Glassy Systems

Glassy systems are characterized by an extremely sluggish dynamics without any simple sign of long range order. It is a debated question whether a correct description of such phenomenon requires the emergence of a large correlation length. We prove rigorous bounds between length and time scales implying the growth of a properly defined length when the relaxation time increases. Our results are valid in a rather general setting, which covers finite-dimensional and mean field systems. As an illustration, we discuss the Glauber (heat bath) dynamics of p-spin glass models on random regular graphs. We present the first proof that a model of this type undergoes a purely dynamical phase transition not accompanied by any thermodynamic singularity.Comment: 24 pages, 3 figures; published versio

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

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 $\phi^4$-Random Field Ising Model. We confirm our analysis studying the finite size scaling of the $p$-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

Solving satisfiability problems by fluctuations: The dynamics of stochastic local search algorithms

Stochastic local search algorithms are frequently used to numerically solve hard combinatorial optimization or decision problems. We give numerical and approximate analytical descriptions of the dynamics of such algorithms applied to random satisfiability problems. We find two different dynamical regimes, depending on the number of constraints per variable: For low constraintness, the problems are solved efficiently, i.e. in linear time. For higher constraintness, the solution times become exponential. We observe that the dynamical behavior is characterized by a fast equilibration and fluctuations around this equilibrium. If the algorithm runs long enough, an exponentially rare fluctuation towards a solution appears.Comment: 21 pages, 18 figures, revised version, to app. in PRE (2003

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