Mixing length scales of low temperature spin plaquettes models

Plaquette models are short range ferromagnetic spin models that play a key role in the dynamic facilitation approach to the liquid glass transition. In this paper we perform a rigorous study of the thermodynamic properties of two dimensional plaquette models, the square and triangular plaquette models. We prove that for any positive temperature both models have a unique infinite volume Gibbs measure with exponentially decaying correlations. We analyse the scaling of three a priori different static correlation lengths in the small temperature regime, the mixing, cavity and multispin correlation lengths. Finally, using the symmetries of the model we determine an exact self similarity property for the infinite volume Gibbs measure.Comment: 33 pages, 9 figure

Relaxation to equilibrium of generalized East processes on Z^d: renormalization group analysis and energy-entropy competition.

We consider a class of kinetically constrained interacting particle systems on Zd which play a key role in several heuristic qualitative and quantitative approaches to describe the complex behavior of glassy dynamics. With rate one and independently among the vertices of Zd, to each occupation variable ηx ∈ {0,1} a new value is pro- posed by tossing a (1 − q)-coin. If a certain local constraint is satisfied by the current configuration the proposed move is accepted, otherwise it is rejected. For d = 1 the con- straint requires that there is a vacancy at the vertex to the left of the updating vertex. In this case the process is the well known East process. On Z2 the West or the South neighbor of the updating vertex must contain a vacancy. Similarly in higher dimen- sions. Despite of their apparent simplicity, in the limit q ↘ 0 of low vacancy density, corresponding to a low temperature physical setting, these processes feature a rather complicated dynamic behavior with hierarchical relaxation time scales, heterogeneity and universality. Using renormalization group ideas, we first show that the relaxation time on Zd scales as the 1/d-root of the relaxation time of the East process, confirming indications coming from massive numerical simulations. Next we compute the relax- ation time in finite boxes by carefully analyzing the subtle energy-entropy competition, using a multi-scale analysis, capacity methods and an algorithmic construction. Our results establish dynamic heterogeneity and a dramatic dependence on the boundary conditions. Finally we prove a rather strong anisotropy property of these processes: the creation of a new vacancy at a vertex x out of an isolated one at the origin (a seed) may occur on (logarithmically) different time scales which heavily depend not only on the l1-norm of x but also on its direction

Cutoff for the East process

The East process is a 1D kinetically constrained interacting particle system, introduced in the physics literature in the early 90's to model liquid-glass transitions. Spectral gap estimates of Aldous and Diaconis in 2002 imply that its mixing time on $L$ sites has order $L$. We complement that result and show cutoff with an $O(\sqrt{L})$-window. The main ingredient is an analysis of the front of the process (its rightmost zero in the setup where zeros facilitate updates to their right). One expects the front to advance as a biased random walk, whose normal fluctuations would imply cutoff with an $O(\sqrt{L})$-window. The law of the process behind the front plays a crucial role: Blondel showed that it converges to an invariant measure $\nu$, on which very little is known. Here we obtain quantitative bounds on the speed of convergence to $\nu$, finding that it is exponentially fast. We then derive that the increments of the front behave as a stationary mixing sequence of random variables, and a Stein-method based argument of Bolthausen ('82) implies a CLT for the location of the front, yielding the cutoff result. Finally, we supplement these results by a study of analogous kinetically constrained models on trees, again establishing cutoff, yet this time with an $O(1)$-window.Comment: 33 pages, 2 figure

Front propagation versus bulk relaxation in the annealing dynamics of a kinetically constrained model of ultrastable glasses

Glasses prepared by physical vapour deposition have been shown to be remarkably more stable than those prepared by standard cooling protocols, with properties that appear to be similar to systems aged for extremely long times. When subjected to a rapid rise in temperature, ultrastable glasses anneal towards the liquid in a qualitatively different manner than ordinary glasses, with the seeming competition of different time and length scales. We numerically reproduce the phenomenology of ultrastable glass annealing with a kinetically constrained model, a three dimensional East model with soft constraints, in a setting where the bulk is in an ultrastable configuration and a free surface is permanently excited. Annealing towards the liquid state is given by the competition between the ballistic propagation of a front from the free surface and a much slower nucleation-like relaxation in the bulk. The crossover between these mechanisms also explains the change in behaviour with film thickness seen experimentally

Condensation in randomly perturbed zero-range processes

The zero-range process is a stochastic interacting particle system that exhibits a condensation transition under certain conditions on the dynamics. It has recently been found that a small perturbation of a generic class of jump rates leads to a drastic change of the phase diagram and prevents condensation in an extended parameter range. We complement this study with rigorous results on a finite critical density and quenched free energy in the thermodynamic limit, as well as quantitative heuristic results for small and large noise which are supported by detailed simulation data. While our new results support the initial findings, they also shed new light on the actual (limited) relevance in large finite systems, which we discuss via fundamental diagrams obtained from exact numerics for finite systems.Comment: 18 pages, 6 figure