Spatial correlations in the relaxation of the Kob-Andersen model

We describe spatio-temporal correlations and heterogeneities in a kinetically constrained glassy model, the Kob-Andersen model. The kinetic constraints of the model alone induce the existence of dynamic correlation lengths, that increase as the density $\rho$ increases, in a way compatible with a double-exponential law. We characterize in detail the trapping time correlation length, the cooperativity length, and the distribution of persistent clusters of particles. This last quantity is related to the typical size of blocked clusters that slow down the dynamics for a given density.Comment: 7 pages, 6 figures, published version (title has changed

Fractional moment bounds and disorder relevance for pinning models

We study the critical point of directed pinning/wetting models with quenched disorder. The distribution K(.) of the location of the first contact of the (free) polymer with the defect line is assumed to be of the form K(n)=n^{-\alpha-1}L(n), with L(.) slowly varying. The model undergoes a (de)-localization phase transition: the free energy (per unit length) is zero in the delocalized phase and positive in the localized phase. For \alpha<1/2 it is known that disorder is irrelevant: quenched and annealed critical points coincide for small disorder, as well as quenched and annealed critical exponents. The same has been proven also for \alpha=1/2, but under the assumption that L(.) diverges sufficiently fast at infinity, an hypothesis that is not satisfied in the (1+1)-dimensional wetting model considered by Forgacs et al. (1986) and Derrida et al. (1992), where L(.) is asymptotically constant. Here we prove that, if 1/21, then quenched and annealed critical points differ whenever disorder is present, and we give the scaling form of their difference for small disorder. In agreement with the so-called Harris criterion, disorder is therefore relevant in this case. In the marginal case \alpha=1/2, under the assumption that L(.) vanishes sufficiently fast at infinity, we prove that the difference between quenched and annealed critical points, which is known to be smaller than any power of the disorder strength, is positive: disorder is marginally relevant. Again, the case considered by Forgacs et al. (1986) and Derrida et al. (1992) is out of our analysis and remains open.Comment: 20 pages, 1 figure; v2: few typos corrected, references revised. To appear on Commun. Math. Phy

Finite size effects in the dynamics of glass-forming liquids

We present a comprehensive theoretical study of finite size effects in the relaxation dynamics of glass-forming liquids. Our analysis is motivated by recent theoretical progress regarding the understanding of relevant correlation length scales in liquids approaching the glass transition. We obtain predictions both from general theoretical arguments and from a variety of specific perspectives: mode-coupling theory, kinetically constrained and defect models, and random first order transition theory. In the latter approach, we predict in particular a non-monotonic evolution of finite size effects across the mode-coupling crossover due to the competition between mode-coupling and activated relaxation. We study the role of competing relaxation mechanisms in giving rise to non-monotonic finite size effects by devising a kinetically constrained model where the proximity to the mode-coupling singularity can be continuously tuned by changing the lattice topology. We use our theoretical findings to interpret the results of extensive molecular dynamics studies of four model liquids with distinct structures and kinetic fragilities. While the less fragile model only displays modest finite size effects, we find a more significant size dependence evolving with temperature for more fragile models, such as Lennard-Jones particles and soft spheres. Finally, for a binary mixture of harmonic spheres we observe the predicted non-monotonic temperature evolution of finite size effects near the fitted mode-coupling singularity, suggesting that the crossover from mode-coupling to activated dynamics is more pronounced for this model. Finally, we discuss the close connection between our results and the recent report of a non-monotonic temperature evolution of a dynamic length scale near the mode-coupling crossover in harmonic spheres.Comment: 19 pages, 10 figures. V2: response to referees + refs added (close to published version

New bounds for the free energy of directed polymers in dimension 1+1 and 1+2

We study the free energy of the directed polymer in random environment in dimension 1+1 and 1+2. For dimension 1, we improve the statement of Comets and Vargas concerning very strong disorder by giving sharp estimates on the free energy at high temperature. In dimension 2, we prove that very strong disorder holds at all temperatures, thus solving a long standing conjecture in the field.Comment: 31 pages, 4 figures, final version, accepted for publication in Communications in Mathematical Physic

Jamming percolation and glassy dynamics

We present a detailed physical analysis of the dynamical glass-jamming transition which occurs for the so called Knight models recently introduced and analyzed in a joint work with D.S.Fisher \cite{letterTBF}. Furthermore, we review some of our previous works on Kinetically Constrained Models. The Knights models correspond to a new class of kinetically constrained models which provide the first example of finite dimensional models with an ideal glass-jamming transition. This is due to the underlying percolation transition of particles which are mutually blocked by the constraints. This jamming percolation has unconventional features: it is discontinuous (i.e. the percolating cluster is compact at the transition) and the typical size of the clusters diverges faster than any power law when $\rho\nearrow\rho_c$. These properties give rise for Knight models to an ergodicity breaking transition at $\rho_c$: at and above $\rho_{c}$ a finite fraction of the system is frozen. In turn, this finite jump in the density of frozen sites leads to a two step relaxation for dynamic correlations in the unjammed phase, analogous to that of glass forming liquids. Also, due to the faster than power law divergence of the dynamical correlation length, relaxation times diverge in a way similar to the Vogel-Fulcher law.Comment: Submitted to the special issue of Journal of Statistical Physics on Spin glasses and related topic

Delocalization transition of the selective interface model: distribution of pseudo-critical temperatures

According to recent progress in the finite size scaling theory of critical disordered systems, the nature of the phase transition is reflected in the distribution of pseudo-critical temperatures $T_c(i,L)$ over the ensemble of samples $(i)$ of size $L$. In this paper, we apply this analysis to the delocalization transition of an heteropolymeric chain at a selective fluid-fluid interface. The width $\Delta T_c(L)$ and the shift $[T_c(\infty)-T_c^{av}(L)]$ are found to decay with the same exponent $L^{-1/\nu_{R}}$, where $1/\nu_{R} \sim 0.26$. The distribution of pseudo-critical temperatures $T_c(i,L)$ is clearly asymmetric, and is well fitted by a generalized Gumbel distribution of parameter $m \sim 3$. We also consider the free energy distribution, which can also be fitted by a generalized Gumbel distribution with a temperature dependent parameter, of order $m \sim 0.7$ in the critical region. Finally, the disorder averaged number of contacts with the interface scales at $T_c$ like $L^{\rho}$ with $\rho \sim 0.26 \sim 1/\nu_R$.Comment: 9 pages,6 figure

Critical properties and finite--size estimates for the depinning transition of directed random polymers

We consider models of directed random polymers interacting with a defect line, which are known to undergo a pinning/depinning (or localization/delocalization) phase transition. We are interested in critical properties and we prove, in particular, finite--size upper bounds on the order parameter (the {\em contact fraction}) in a window around the critical point, shrinking with the system size. Moreover, we derive a new inequality relating the free energy \tf and an annealed exponent $\mu$ which describes extreme fluctuations of the polymer in the localized region. For the particular case of a $(1+1)$--dimensional interface wetting model, we show that this implies an inequality between the critical exponents which govern the divergence of the disorder--averaged correlation length and of the typical one. Our results are based on on the recently proven smoothness property of the depinning transition in presence of quenched disorder and on concentration of measure ideas.Comment: 15 pages, 1 figure; accepted for publication on J. Stat. Phy

Statistics of Microstructure Formation in Martensitic Transitions Studied by a Random-Field Potts Model with Dipolar-like Interactions

We have developed a simple model for the study of a cubic to tetragonal martensitic transition, under athermal conditions, in systems with a certain amount of disorder. We have performed numerical simulations that allow for a statistical study of the dynamics of the transition when the system is driven from the high-temperature cubic phase to the low-temperature degenerate tetragonal phase. Our goal is to reveal the existence of kinetic constraints that arise from competition between the equivalent variants of the martensitic phase, and which prevent the system from reaching optimal final microstructures.Comment: 11 pages, 14 figure

Group testing with Random Pools: Phase Transitions and Optimal Strategy

The problem of Group Testing is to identify defective items out of a set of objects by means of pool queries of the form "Does the pool contain at least a defective?". The aim is of course to perform detection with the fewest possible queries, a problem which has relevant practical applications in different fields including molecular biology and computer science. Here we study GT in the probabilistic setting focusing on the regime of small defective probability and large number of objects, $p \to 0$ and $N \to \infty$. We construct and analyze one-stage algorithms for which we establish the occurrence of a non-detection/detection phase transition resulting in a sharp threshold, $\bar M$, for the number of tests. By optimizing the pool design we construct algorithms whose detection threshold follows the optimal scaling $\bar M\propto Np|\log p|$. Then we consider two-stages algorithms and analyze their performance for different choices of the first stage pools. In particular, via a proper random choice of the pools, we construct algorithms which attain the optimal value (previously determined in Ref. [16]) for the mean number of tests required for complete detection. We finally discuss the optimal pool design in the case of finite $p$

Cooperative Behavior of Kinetically Constrained Lattice Gas Models of Glassy Dynamics

Kinetically constrained lattice models of glasses introduced by Kob and Andersen (KA) are analyzed. It is proved that only two behaviors are possible on hypercubic lattices: either ergodicity at all densities or trivial non-ergodicity, depending on the constraint parameter and the dimensionality. But in the ergodic cases, the dynamics is shown to be intrinsically cooperative at high densities giving rise to glassy dynamics as observed in simulations. The cooperativity is characterized by two length scales whose behavior controls finite-size effects: these are essential for interpreting simulations. In contrast to hypercubic lattices, on Bethe lattices KA models undergo a dynamical (jamming) phase transition at a critical density: this is characterized by diverging time and length scales and a discontinuous jump in the long-time limit of the density autocorrelation function. By analyzing generalized Bethe lattices (with loops) that interpolate between hypercubic lattices and standard Bethe lattices, the crossover between the dynamical transition that exists on these lattices and its absence in the hypercubic lattice limit is explored. Contact with earlier results are made via analysis of the related Fredrickson-Andersen models, followed by brief discussions of universality, of other approaches to glass transitions, and of some issues relevant for experiments.Comment: 59 page