Maria Eulalia Vares

Large deviations and metastability

Preface 1. Large deviations: basic results 2. Small random perturbations of dynamical systems: basic estimates of Freidlin and Wentzell 3. Large deviations and statistical mechanics 4. Metastability I: general description, the Curie-Weiss model and contact processes 5. Metastability II: the models of Freidlin and Wentzell 6. Reversible Markov chains in the Freidlin-Wentzell regime 7. Metastable behaviour for lattice spin models at low temperature Bibliography Index.

Metastable behavior of stochastic dynamics: A pathwise approach

In this paper a new approach to metastability for stochastic dynamics is proposed. The basic idea is to study the statistics of each path, performing time averages along the evolution. Metastability would be characterized by the fact that the process of these time averages converges, under a suitable rescaling, to a measure valued Markov jump process. Here this convergence is shown for the Curie-Weiss mean field dynamics and also for a model with spatial structure: Harris contact process.

Hydrodynamic equations for attractive particle systems on Z

Hydrodynamic properties for a class of nondiffusive particle systems are investigated. The method allows one to study local equilibria for a class of asymmetric zero-range processes, and applies as well to other models, such as asymmetric simple exclusion and “misanthropes.” Attractiveness is an essential ingredient. The hydrodynamic equations present shock wave phenomena. Preservation of local equilibrium is proven to hold away from the shocks. The problem of breakdown of local ergodicity at the shocks, which was investigated by D. Wick in a particular model, remains open in this more general setup.

Large deviations for a reaction diffusion model

SummaryWe obtain large deviation estimates for the empirical measure of a class of interacting particle systems. These consist of a superposition of Glauber and Kawasaki dynamics and are described, in the hydrodynamic limit, by a reaction diffusion equation. We extend results of Kipnis, Olla and Varadhan for the symmetric exclusion process, and provide an approximation scheme for the rate functional. Some physical implications of our results are briefly indicated.

The asymmetric simple exclusion model with multiple shocks

We consider the one-dimensional totally asymmetric simple exclusion process with initial product distribution with densities 0≤ρ0<ρ1<⋯<ρn≤1 in (−∞,c1e−1), [c1e−1,c2e−1),…,[cne−1,+∞), respectively. The initial distribution has shocks (discontinuities) at e−1ck, k=1,…,n, and we assume that in the corresponding macroscopic Burgers equation the n shocks meet in r∗ at time t∗. The microscopic position of the shocks is represented by second class particles whose distribution in the scale e−1/2 is shown to converge to a function of n independent Gaussian random variables representing the fluctuations of these particles “just before the meeting”. We show that the density field at time e−1t∗, in the scale e−1/2 and as seen from e−1r∗ converges weakly to a random measure with piecewise constant density as e→0; the points of discontinuity depend on these limiting Gaussian variables. As a corollary we show that, as e→0, the distribution of the process at site e−1r∗+e−1/2a at time e−1t∗ tends to a non-trivial convex combination of the product measures with densities ρk, the weights of the combination being explicitly computable.

The symmetric simple exclusion process, I: Probability estimates

We consider the one-dimensional symmetric simple exclusion process with nearest neighbor jumps and we prove estimates on the decay of the correlation functions at long times.

Transient bimodality in interacting particle systems

We consider a system of spins which have values ±1 and evolve according to a jump Markov process whose generator is the sum of two generators, one describing a spin-flipGlauber process, the other aKawasaki (stirring) evolution. It was proven elsewhere that if the Kawasaki dynamics is speeded up by a factor ε−2, then, in the limit ε → 0 (continuum limit), propagation of chaos holds and the local magnetization solves a reaction-diffusion equation. We choose the parameters of the Glauber interaction so that the potential of the reaction term in the reaction-diffusion equation is a double-well potential with quartic maximum at the origin. We assume further that for each ε the system is in a finite interval ofZ with ε−1 sites and periodic boundary conditions. We specify the initial measure as the product measure with 0 spin average, thus obtaining, in the continuum limit, a constant magnetic profile equal to 0, which is a stationary unstable solution to the reaction-diffusion equation. We prove that at times of the order ε−1/2 propagation of chaos does not hold any more and, in the limit as ε → 0, the state becomes a nontrivial superposition of Bernoulli measures with parameters corresponding to the minima of the reaction potential. The coefficients of such a superposition depend on time (on the scale ε−1/2) and at large times (on this scale) the coefficient of the term corresponding to the initial magnetization vanishes (transient bimodality). This differs from what was observed by De Masi, Presutti, and Vares, who considered a reaction potential with quadratic maximum and no bimodal effect was seen, as predicted by Broggi, Lugiato, and Colombo.

The survival of the large dimensional basic contact process

SummaryWe consider the d-dimensional basic contact process obtaining the limit value of the probability of survival when d→+∞, and showing that the finite dimensional distributions of the upper invariant measure become of the product form as d→+∞.

On the truncated anisotropic long-range percolation on Z2

Consider the following bond percolation process on : each vertex is connected to each of its nearest neighbour in the vertical direction with probability pv=[var epsilon]>0; and in the horizontal direction each vertex is connected to each of the vertices x±(i,0) with probability pi[greater-or-equal, slanted]0, i[greater-or-equal, slanted]1, with all different connections being independent. We prove that if pis satisfy some regularity property, namely if pi[greater-or-equal, slanted]1/i ln i, for i sufficiently large, then for each [var epsilon]>0 there exists K[reverse not equivalent]K([var epsilon]) such that for truncated percolation process (for which if i[less-than-or-equals, slant]K and if j>K) the probability of the open cluster of the origin to be infinite remains positive.

Random spatial growth with paralyzing obstacles.

We study models of spatial growth processes where initially there are sources of growth (indicated by the colour green) and sources of a growth-stopping (paralyzing) substance (indicated by red). The green sources expand and may merge with others (there is no ‘inter-green’ competition). The red substance remains passive as long as it is isolated. However, when a green cluster comes in touch with the red substance, it is immediately invaded by the latter, stops growing and starts to act as red substance itself. In our main model space is represented by a graph, of which initially each vertex is randomly green, red or white (vacant), and the growth of the green clusters is similar to that in first-passage percolation. The main issues we investigate are whether the model is well-defined on an infinite graph (e.g. the d-dimensional cubic lattice), and what can be said about the distribution of the size of a green cluster just before it is paralyzed. We show that, if the initial density of red vertices is positive, and that of white vertices is sufficiently small, the model is indeed well-defined and the above distribution has an exponential tail. In fact, we believe this to be true whenever the initial density of red is positive. This research also led to a relation between invasion percolation and critical Bernoulli percolation which seems to be of independent interest. Research funded in part by the Dutch BSIK/BRICKS project. Research supported in part by NSF grant DMS-0605166. Partially supported by CNPq, Brazil Partially supported by CNPq, Brazil