We describe the motion of a tracer in an environment given by a kinetically constrained spin model (KCSM) at equilibrium. We check convergence of its trajectory properly rescaled to a Brownian motion and positivity of the diffusion coefficient $D$ as soon as the spectral gap of the environment is positive (which coincides with the ergodicity region under general conditions). Then we study the asymptotic behavior of $D$ when the density $1-q$ of the environment goes to $1$ in two classes of KCSM. For noncooperative models, the diffusion coefficient $D$ scales like a power of $q$, with an exponent that we compute explicitly. In the case of the Fredrickson-Andersen one-spin facilitated model, this proves a prediction made in Jung, Garrahan and Chandler [Phys. Rev. E 69 (2004) 061205]. For the East model, instead we prove that the diffusion coefficient is comparable to the spectral gap, which goes to zero faster than any power of $q$. This result contradicts the prediction of physicists (Jung, Garrahan and Chandler [Phys. Rev. E 69 (2004) 061205; J. Chem. Phys. 123 (2005) 084509]), based on numerical simulations, that suggested $D\sim \operatorname {gap}^ξ$ with $ξ<1$.
Kinetically constrained lattice gases (KCLG) are interacting particle systems on the integer lattice $\mathbb Z^d$ with hard core exclusion and Kawasaki type dynamics. Their peculiarity is that jumps are allowed only if the configuration satisfies a constraint which asks for enough empty sites in a certain local neighborhood. KCLG have been introduced and extensively studied in physics literature as models of glassy dynamics. We focus on the most studied class of KCLG, the Kob Andersen (KA) models. We analyze the behavior of a tracer (i.e. a tagged particle) at equilibrium. We prove that for a…
We study a family of conservative interacting particle systems with degenerate rates called noncooperative kinetically constrained lattice gases. We prove for all models in this family the diffusive scaling of the relaxation time, the positivity of the diffusion coefficient, and the positivity of the self-diffusion coefficient.
We study two kinetically constrained models in a quenched random environment. The first model is a mixed threshold Fredrickson-Andersen model on $\mathbb{Z}^{2}$, where the update threshold is either $1$ or $2$. The second is a mixture of the Fredrickson-Andersen $1$-spin facilitated constraint and the North-East constraint in $\mathbb{Z}^{2}$. We compare three time scales related to these models - the bootstrap percolation time for emptying the origin, the relaxation time of the kinetically constrained model, and the time for emptying the origin of the kinetically constrained model - and unde…
In the hydrodynamic regime, the evolution of a stochastic lattice gas with symmetric hopping rules is described by a diffusion equation with density-dependent diffusion coefficient encapsulating all microscopic details of the dynamics. This diffusion coefficient is, in principle, determined by a Green-Kubo formula. In practice, even when the equilibrium properties of a lattice gas are analytically known, the diffusion coefficient cannot be computed except when a lattice gas additionally satisfies the gradient condition. We develop a procedure to systematically obtain analytical approximations …
Recent years have seen a great deal of progress in our understanding of bootstrap percolation models, a particular class of monotone cellular automata. In the two dimensional lattice there is now a quite satisfactory understanding of their evolution starting from a random initial condition, with a strikingly beautiful universality picture for their critical behaviour. Much less is known for their non-monotone stochastic counterpart, namely kinetically constrained models (KCM). In KCM each vertex is resampled (independently) at rate one by tossing a p-coin iff it can be infected in the next ste…
We analyze the mixing time of a class of oriented kinetically constrained spin models (KCMs) on a d-dimensional lattice of $n^d$ sites. A typical example is the North-East model, a 0-1 spin system on the two-dimensional integer lattice that evolves according to the following rule: whenever a site's southerly and westerly nearest neighbours have spin 0, with rate one it resets its own spin by tossing a p-coin, at all other times its spin remains frozen. Such models are very popular in statistical physics because, in spite of their simplicity, they display some of the key features of the dynamic…
We study the set of possible configurations for a general kinetically constrained model (KCM), a non monotone version of the $\mathcal{U}$-bootstrap percolation cellular automata. We solve a combinatorial question that is a generalization of a problem addressed by Chung, Diaconis and Graham in 2001 for a specific one-dimensional KCM, the East model. Since the general models we consider are in any dimension and lack the oriented character of the East dynamics, we have to follow a completely different route than the one taken by Chung, Diaconis and Graham. Our combinatorial result is used by Mar…
The bisection method for kinetically constrained models (KCM) of Cancrini, Martinelli, Roberto and Toninelli is a vital technique applied also beyond KCM. In this note we present a new way of performing it, based on a novel two-block dynamics with a probabilistic proof instead of the original spectral one. We illustrate the method by very directly proving an upper bound on the relaxation time of KCM like the one for the East model in a strikingly general setting. Namely, we treat KCM on finite or infinite one-dimensional volumes, with any boundary condition, conditioned on any of the irreducib…
We analyze the density and size dependence of the relaxation time for kinetically constrained spin models (KCSM) intensively studied in the physical literature as simple models sharing some of the features of a 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 of empty sites around it satisfies certain constraints which completely define each specifi…
On the rooted $k$-ary tree we consider a 0-1 kinetically constrained spin model in which the occupancy variable at each node is re-sampled with rate one from the Bernoulli(p) measure iff all its children are empty. For this process the following picture was conjectured to hold. As long as $p$ is below the percolation threshold $p_c=1/k$ the process is ergodic with a finite relaxation time while, for $p>p_c$, the process on the infinite tree is no longer ergodic and the relaxation time on a finite regular sub-tree becomes exponentially large in the depth of the tree. At the critical point $p=p_…
We analyze kinetically constrained 0-1 spin models (KCSM) on rooted and unrooted trees of finite connectivity. We focus in particular on the class of Friedrickson-Andersen models FA-jf and on an oriented version of them. These tree models are particularly relevant in physics literature since some of them undergo an ergodicity breaking transition with the mixed first-second order character of the glass transition. Here we first identify the ergodicity regime and prove that the critical density for FA-jf and OFA-jf models coincide with that of a suitable bootstrap percolation model. Next we prov…
Kinetically constrained models (KCM) are reversible interacting particle systems on $\mathbb Z^d$ with continuous time Markov dynamics of Glauber type, which represent a natural stochastic (and non-monotone) counterpart of the family of cellular automata known as $\mathcal U$-bootstrap percolation. Furthermore, KCM have an interest in their own since they display some of the most striking features of the liquid-glass transition, a major and longstanding open problem in condensed matter physics. A key issue for KCM is to identify the scaling of the characteristic time scales when the equilibriu…
Kinetically constrained models (KCM) are reversible interacting particle systems on $\mathbb Z^d$ with continuous time Markov dynamics of Glauber type, which represent a natural stochastic (and non-monotone) counterpart of the family of cellular automata known as $\mathcal U$-bootstrap percolation. KCM also display some of the peculiar features of the so-called "glassy dynamics", and as such they are extensively used in the physics literature to model the liquid-glass transition, a major and longstanding open problem in condensed matter physics. We consider two-dimensional KCM with update rule…
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…
We study the Kinetically Constrained Model on the polluted square lattice, with two-neighbor constraints. For a quenched polluted environment with low pollution density we give bounds on the infection time of the origin.
The Kob-Andersen model is a fundamental example of a kinetically constrained lattice gas, that is, an interacting particle system with Kawasaki type dynamics and kinetic constraints. In this model, a particle is allowed to jump when sufficiently many neighboring sites are empty. We study the motion of a single tagged particle and in particular its convergence to a Brownian motion. Previous results showed that the path of this particle indeed converges in diffusive time-scale, and the purpose of this paper is to study the rate of decay of the self-diffusion coefficient for large densities. We f…
We study a kinetically constrained Ising process (KCIP) associated with a graph $G$ and density parameter $p$; this process is an interacting particle system with state space $\{ 0, 1 \}^{G}$, the location of the particles. The `constraint' in the name of the process refers to the rule that a vertex cannot change its state unless it has at least one neighbour in state `1'. The KCIP has been proposed by statistical physicists as a model for the glass transition. In this note, we study the mixing time of a KCIP on the 2-dimensional torus $G = \mathbb{Z}_{L}^{2}$ in the low-density regime $p = \f…
We study a kinetically constrained Ising process (KCIP) associated with a graph G and density parameter p; this process is an interacting particle system with state space $\{0,1\}^{G}$. The stationary distribution of the KCIP Markov chain is the Binomial($|G|, p$) distribution on the number of particles, conditioned on having at least one particle. The `constraint' in the name of the process refers to the rule that a vertex cannot change its state unless it has at least one neighbour in state `1'. The KCIP has been proposed by statistical physicists as a model for the glass transition, and mor…
