Toninelli, Biroli, and Fisher Reply: On the universality of jamming percolation. -In Ref.[1], we introduced a class of kinetically constrained models which display a dynamical glass transition: Above a critical density c , there appears an infinite cluster of forever frozen particles. At c , the density of frozen particles is discontinuous, while as % c there is an exponentially diverging crossover length. This jamming percolation behavior in two dimensions is a consequence of two perpendicular directedpercolation (DP)-like processes which together can form a frozen network of DP segments ending at T junctions with perpendicular DP segments.In Ref.[1], we focused on a particular example: the ''knights'' model. As correctly pointed out by Jeng and Schwarz (JS) [2], we overlooked some frozen structures which are not simple DP paths: These ''thicker'' directed structures lower the critical density. Here we argue that, nevertheless, the full directed processes are in the DP universality class, and the T junctions between these give rise to a jamming percolation transition with the same universal properties [1]. Moreover, for a simpler model our results are rigorous. The ''spiral'' model is similar to the knights model except that blocking of a particle is by either its N, NE and S, SW or its W, NW and E, SE pairs of neighbors as in Fig. 1(b) [cf. Fig. 1(b) in Ref. [1]]. There are two directed processes, in the NNE-SSW and the ESE-WNW directions, which are congruent to DP processes on a square lattice. As infinite occupied DP paths are here necessary and sufficient for an infinite frozen structure, c DP c and our results are rigorous [3]. We claim that the knights and spiral models are in the same universality class. In each diagonal direction of the knights model, there are two infinite sequences of thicker and thicker DP processes, SDP and NDP, for which the

We compute analytically and numerically the four-point correlation function that characterizes non-trivial cooperative dynamics in glassy systems within several models of glasses: elasto-plastic deformations, mode-coupling theory (MCT), collectively rearranging regions (CRR), diffusing defects and kinetically constrained models (KCM). Some features of the four-point susceptibility χ4(t) are expected to be universal: at short times we expect a power-law increase in time as t 4 due to ballistic motion (t 2 if the dynamics is Brownian) followed by an elastic regime (most relevant deep in the glass phase) characterized by a t or √ t growth, depending on whether phonons are propagative or diffusive. We find both in the β, and the early α regime that χ4 ∼ t µ , where µ is directly related to the mechanism responsible for relaxation. This regime ends when a maximum of χ4 is reached at a time t = t * of the order of the relaxation time of the system. This maximum is followed by a fast decay to zero at large times. The height of the maximum also follows a power-law, χ4(t * ) ∼ t * λ . The value of the exponents µ and λ allows one to distinguish between different mechanisms. For example, freely diffusing defects in d = 3 lead to µ = 2 and λ = 1, whereas the CRR scenario rather predicts either µ = 1 or a logarithmic behaviour depending on the nature of the nucleation events, and a logarithmic behaviour of χ4(t * ). MCT leads to µ = b and λ = 1/γ, where b and γ are the standard MCT exponents. We compare our theoretical results with numerical simulations on a Lennard-Jones and a soft-sphere system. Within the limited time-scales accessible to numerical simulations, we find that the exponent µ is rather small, µ < 1, with a value in reasonable agreement with the MCT predictions, but not with the prediction of simple diffusive defect models, KCMs with non-cooperative defects and CRR. Experimental and numerical determination of χ4(t) for longer time scales and lower temperatures would yield highly valuable information on the glass formation mechanism.

We analyze the density and size dependence of the relaxation time for kinetically constrained spin models (KCSM) intensively studied in the physics literature as simple models sharing some of the features of the glass transition. KCSM are interacting particle systems on Z d with Glauber-like dynamics, reversible w.r.t. a simple product i.i.d Bernoulli(p) measure. The essential feature of a KCSM is that the creation/destruction of a particle at a given site can occur only if the current configuration around it satisfies certain constraints which completely define each specific model. No other interaction is present in the model. From the mathematical point of view, the basic issues concerning positivity of the spectral gap inside the ergodicity region and its scaling with the particle density p remained open for most KCSM (with the notably exception of the East model in d = 1; Aldous and Diaconis in J Stat Phys 107(5-6):945-975, 2002). Here for the first time we: (i) identify the ergodicity region by establishing a connection with an associated bootstrap percolation model; 460 N. Cancrini et al.(ii) develop a novel multi-scale approach which proves positivity of the spectral gap in the whole ergodic region; (iii) establish, sometimes optimal, bounds on the behavior of the spectral gap near the boundary of the ergodicity region and (iv) establish pure exponential decay at equilibrium for the persistence function, i.e. the probability that the occupation variable at the origin does not change before time t. Our techniques are flexible enough to allow a variety of constraints and our findings disprove certain conjectures which appeared in the physical literature on the basis of numerical simulations.

Abstract. -We show that facilitated spin models of cooperative dynamics introduced by Fredrickson and Andersen display on Bethe lattices a glassy behaviour similar to the one predicted by the mode-coupling theory of supercooled liquids and the dynamical theory of mean-field disordered systems. At low temperature such cooperative models show a two-step relaxation and their equilibration time diverges at a finite temperature according to a powerlaw. The geometric nature of the dynamical arrest corresponds to a bootstrap percolation process which leads to a phase space organization similar to the one of mean-field disordered systems. The relaxation dynamics after a subcritical quench exhibits aging and converges asymptotically to the threshold states that appear at the bootstrap percolation transition.Introduction. -Lattice models are widely used in statistical mechanics to gain a qualitative and often deeper understanding of physical phenomena. In a seminal work Fredrickson and Andersen (FA) [1] introduced a simple lattice spin model of the liquid-glass transition, whose Hamiltonian corresponds to uncoupled Ising spins in a positive magnetic field [2,3]. The spins represent a coarse-grained region of the liquid with high (−1 spins) or low (+1 spins) mobility and the magnetic field, that favors up spins, leads to very few mobile regions embedded in an immobile background at low temperature. The spin dynamics is subjected to a kinetic constraint: for each time step a randomly selected spin can flip only if the number of nearest neighbor down spins is larger or equal than f , where the facilitation parameter f is a number between zero and the lattice connectivity. The kinetic constraint mimics at a coarse grained level the cage effect in super-cooled liquids, where particles rattle in the cage formed by their neighbors and then move further if they are able to find a way through the surrounding particles. At low temperature/high density the latter process is strongly inhibited leading to an arrest of particle motion over macroscopic time scales. Similarly, at high temperature the kinetic constraint plays a little role and the relaxation is fast, whereas at low temperature