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

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

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

Magnetization enumerator of real-valued symmetric channels in Gallager error-correcting codes

Using the magnetization enumerator method, we evaluate the practical and theoretical limitations of symmetric channels with real outputs. Results are presented for several regular Gallager code constructions.Comment: 5 pages, 1 figure, to appear as Brief Report in Physical Review

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

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

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

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

Avalanches in mean-field models and the Barkhausen noise in spin-glasses

We obtain a general formula for the distribution of sizes of "static avalanches", or shocks, in generic mean-field glasses with replica-symmetry-breaking saddle points. For the Sherrington-Kirkpatrick (SK) spin-glass it yields the density rho(S) of the sizes of magnetization jumps S along the equilibrium magnetization curve at zero temperature. Continuous replica-symmetry breaking allows for a power-law behavior rho(S) ~ 1/(S)^tau with exponent tau=1 for SK, related to the criticality (marginal stability) of the spin-glass phase. All scales of the ultrametric phase space are implicated in jump events. Similar results are obtained for the sizes S of static jumps of pinned elastic systems, or of shocks in Burgers turbulence in large dimension. In all cases with a one-step solution, rho(S) ~ S exp(-A S^2). A simple interpretation relating droplets to shocks, and a scaling theory for the equilibrium analog of Barkhausen noise in finite-dimensional spin glasses are discussed.Comment: 6 pages, 1 figur