Propagation of Chaos for a Balls into Bins Model

Consider a finite number of balls initially placed in $L$ bins. At each time step a ball is taken from each non-empty bin. Then all the balls are uniformly reassigned into bins. This finite Markov chain is called Repeated Balls-into-Bins process and is a discrete time interacting particle system with parallel updating. We prove that, starting from a suitable (chaotic) set of initial states, as $L\to+\infty$, the numbers of balls in each bin becomes independent from the rest of the system i.e. we have propagation of chaos. We furthermore study some equilibrium properties of the limiting nonlinear process

Kinetically constrained spin models

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 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 $d=1$ \cite{Aldous-Diaconis}). Here for the first time we: i) identify the ergodicity region by establishing a connection with an associated bootstrap percolation model; 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 for the persistence function. 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

Propagation of chaos for a General Balls into Bins dynamics

Consider $N$ balls initially placed in $L$ bins. At each time step take a ball from each non-empty bin and \emph{randomly} reassign the balls into the bins.We call this finite Markov chain \emph{General Repeated Balls into Bins} process. It is a discrete time interacting particles system with parallel updates. Assuming a \emph{quantitative} chaotic condition on the reassignment rule we prove a \emph{quantitative} propagation of chaos for this model. We furthermore study some equilibrium properties of the limiting nonlinear process

Mixing time of a kinetically constrained spin model on trees: power law scaling at criticality

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 pc = 1/k the process is ergodic with a finite relaxation time while, for p > pc, 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 = pc the process on the infinite tree is still ergodic but with an infinite relaxation time. Moreover, on finite sub-trees, the relaxation time grows polynomially in the depth of the tree. The conjecture was recently proved by the second and forth author except at crit- icality. Here we analyse the critical and quasi-critical case and prove for the relevant time scales: (i) power law behaviour in the depth of the tree at p = pc and (ii) power law scaling in (pc − p)−1 when p approaches pc from below. Our results, which are very close to those obtained recently for the Ising model at the spin glass critical point, represent the first rigorous analysis of a kinetically constrained model at criticality

Facilitated oriented spin models:some non equilibrium results

We analyze the relaxation to equilibrium for kinetically constrained spin models (KCSM) when the initial distribution $\nu$ is different from the reversible one, $\mu$. This setting has been intensively studied in the physics literature to analyze the slow dynamics which follows a sudden quench from the liquid to the glass phase. We concentrate on two basic oriented KCSM: the East model on \bbZ, for which the constraint requires that the East neighbor of the to-be-update vertex is vacant and the model on the binary tree introduced in \cite{Aldous:2002p1074}, for which the constraint requires the two children to be vacant. While the former model is ergodic at any $p\neq 1$, the latter displays an ergodicity breaking transition at $p_c=1/2$. For the East we prove exponential convergence to equilibrium with rate depending on the spectral gap if $\nu$ is concentrated on any configuration which does not contain a forever blocked site or if $\nu$ is a Bernoulli($p'$) product measure for any $p'\neq 1$. For the model on the binary tree we prove similar results in the regime $p,p'<p_c$ and under the (plausible) assumption that the spectral gap is positive for $p<p_c$. By constructing a proper test function we also prove that if $p'>p_c$ and $p\leq p_c$ convergence to equilibrium cannot occur for all local functions. Finally we present a very simple argument (different from the one in \cite{Aldous:2002p1074}) based on a combination of combinatorial results and ``energy barrier'' considerations, which yields the sharp upper bound for the spectral gap of East when $p\uparrow 1$.Comment: 16 page

On the dynamical behavior of the ABC model

We consider the ABC dynamics, with equal density of the three species, on the discrete ring with $N$ sites. In this case, the process is reversible with respect to a Gibbs measure with a mean field interaction that undergoes a second order phase transition. We analyze the relaxation time of the dynamics and show that at high temperature it grows at most as $N^2$ while it grows at least as $N^3$ at low temperature

Relaxation times of kinetically constrained spin models with glassy dynamics

We analyze the density and size dependence of the relaxation time $\tau$ for kinetically constrained spin systems. These have been proposed as models for strong or fragile glasses and for systems undergoing jamming transitions. For the one (FA1f) or two (FA2f) spin facilitated Fredrickson-Andersen model at any density $\rho<1$ and for the Knight model below the critical density at which the glass transition occurs, we show that the persistence and the spin-spin time auto-correlation functions decay exponentially. This excludes the stretched exponential relaxation which was derived by numerical simulations. For FA2f in $d\geq 2$, we also prove a super-Arrhenius scaling of the form $\exp(1/(1-\rho))\leq \tau\leq\exp(1/(1-\rho)^2)$. For FA1f in $d$=$1,2$ we rigorously prove the power law scalings recently derived in \cite{JMS} while in $d\geq 3$ we obtain upper and lower bounds consistent with findings therein. Our results are based on a novel multi-scale approach which allows to analyze $\tau$ in presence of kinetic constraints and to connect time-scales and dynamical heterogeneities. The techniques are flexible enough to allow a variety of constraints and can also be applied to conservative stochastic lattice gases in presence of kinetic constraints.Comment: 4 page