40 pages, 1 figure. Published in at http://dx.doi.org/10.1214/12-AOP762 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)International audienceWe consider a cluster growth model on the d-dimensional lattice, called internal diffusion limited aggregation (internal DLA). In this model, random walks start at the origin, one at a time, and stop moving when reaching a site not occupied by previous walks. It is known that the asymptotic shape of the cluster is spherical. When dimension is 2 or more, we prove that fluctuations with respect to a sphere are at most a power of the logarithm of its radius in dimension d larger than or equal to 2. In so doing, we introduce a closely related cluster growth model, that we call the flashing process, whose fluctuations are controlled easily and accurately. This process is coupled to internal DLA to yield the desired bound. Part of our proof adapts the approach of Lawler, Bramson and Griffeath, on another space scale, and uses a sharp estimate (written by Blachère in our Appendix) on the expected time spent by a random walk inside an annulus
Abstract. We extend the Dirichlet principle to non-reversible Markov processes on countable state spaces. We present two variational formulas for the solution of the Poisson equation or, equivalently, for the capacity between two disjoint sets. As an application we prove a some recurrence theorems. In particular, we show the recurrence of two-dimensional cycle random walks under a second moment condition on the winding numbers.
ABSTRACT. We establish metastability in the sense of Lebowitz and Penrose under practical and simple hypotheses for Markov chains on a finite configuration space in some asymptotic regime. By comparing restricted ensemble and quasi-stationary measures, and introducing soft measures as interpolation between the two, we prove asymptotic exponential exit law and, on a generally different time scale, asymptotic exponential transition law. By using potential-theoretic tools, and introducing "(κ, λ)-capacities", we give sharp estimates on relaxation time, as well as mean exit time and transition time. We also establish local thermalization on shorter time scales. (a) only one thermodynamic phase is present, (b) a system that starts in this state is likely to take a long time to get out, (c) once the system has gotten out, it is unlikely to return. We can think, for example, about freezing fog made of small droplets in which only one phase is present (liquid phase) that remains for a long time in such a state (until collision with ground or trees, forming then hard rime) and that once frozen will typically not return to liquid state before pressure or temperature have changed.To model such a state they considered in [4] a deterministic dynamics with equilibrium measure µ. First, they associated with the metastable phase a subset R of the configuration space, and described this metastable state by the restricted ensemble µ R = µ(·|R). Second, they proved that the escape rate from R of the system started in µ R is maximal at time t = 0, and that this initial escape rate is very small. Last, they used standard methods of equilibrium statistical mechanics to deal with (c). As an estimate of the returning probability to the metastable state they used the fraction of members of the equilibrium ensemble that have configurations in R and they noted ([4], Section 8):This amounts to assuming that a system whose dynamical state has just left R is no more likely to return to it than one whose dynamical state was never anywhere near R. The validity of this assumption, at least in the short run, is dubious, but at least it provides us with some indication of what to expect. In this paper we want to give a different model for the same phenomenology that overcomes the last difficulty. We will work with stochastic processes rather than deterministic dynamics, but the Lebowitz-Penrose modelization will be our guideline. We will try to recover this phenomenology under simple and practical hypotheses only. Since the study of metastability has been considerably enriched after the Lebowitz and Penrose work, we want also to incorporate in our modeling as much as possible of what was previously achieved. We will then make a brief and partial review of these achievements. Our goals and starting ideas will depend on this review but not our proofs, since we want to make this paper as self-contained as possible. Our model and results are presented in Section 2. Examples of applications are given in Section 7.
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