We prove a strong law of large numbers and an annealed invariance principle for a random walk in a one-dimensional dynamic random environment evolving as the simple exclusion process with jump parameter γ. First, we establish that if the asymptotic velocity of the walker is non-zero in the limiting case "γ = ∞", where the environment gets fully refreshed between each step of the walker, then, for γ large enough, the walker still has a non-zero asymptotic velocity in the same direction.Second, we establish that if the walker is transient in the limiting case γ = 0, then, for γ small enough but positive, the walker has a non-zero asymptotic velocity in the direction of the transience. These two limiting velocities can sometimes be of opposite sign. In all cases, we show that the fluctuations are normal. IntroductionThe question of the evolution of a random walk in a disordered environment has attracted a lot of attention in both the mathematical and the physical communities over the past few decades. The first studies were concerned with static random environments. In this set-up, anomalous slowdowns are expected in comparison with the homogeneous case, as the environment may create traps where the walker gets stuck for long times. This effect is particularly strong in one dimension, where it is by now well understood (see [24] for background). In dynamical random environments instead, the transition probabilities of the walker evolve with time too. If the environment has good space-time mixing properties, one expects the trapping phenomenon to disappear, and the walker to behave very much as if the medium was homogeneous. The study of this case has recently led to intense researches; see for example [13,21] and references therein, as well as [2] for an overview and further references.However, examples of dynamical environments with slow relaxation times occur naturally. Indeed, in the presence of a macroscopically conserved quantity, the environment may evolve diffusively. Time correlations then only decay as t −d/2 in dimension d. Especially for d = 1, correlations decay so slowly that the results developed for fast mixing environments do not apply (see e.g. [3,10,13,20,21]). Slowly mixing environments form an intermediate class of models, which is still far for being well understood at the present time [2,5].

Abstract. We consider the Glauber dynamics for the Ising model with "+" boundary conditions, at zero temperature or at temperature which goes to zero with the system size (hence the quotation marks in the title). In dimension d = 3 we prove that an initial domain of linear size L of "−" spins disappears within a time τ+ which is at most L 2 (log L) c and at least L 2 /(c log L), for some c > 0. The proof of the upper bound proceeds via comparison with an auxiliary dynamics which mimics the motion by mean curvature that is expected to describe, on large time-scales, the evolution of the interface between "+" and "−" domains. The analysis of the auxiliary dynamics requires recent results on the fluctuations of the height function associated to dimer coverings of the infinite honeycomb lattice. Our result, apart from the spurious logarithmic factor, is the first rigorous confirmation of the expected behavior τ+ ≃ const × L 2 , conjectured on heuristic grounds [13,7]. In dimension d = 2, τ+ can be shown to be of order L 2 without logarithmic corrections: the upper bound was proven in [8] and here we provide the lower bound. For d = 2, we also prove that the spectral gap of the generator behaves like c/L for L large, as conjectured in [3].

Abstract. Let D be a simply connected, smooth enough domain of R 2 . For L > 0 consider the continuous time, zero-temperature heat bath dynamics for the nearest-neighbor Ising model on Z 2 with initial condition such that σx = −1 if x ∈ LD and σx = +1 otherwise. It is conjectured [24] that, in the diffusive limit where space is rescaled by L, time by L 2 and L → ∞, the boundary of the droplet of "−" spins follows a deterministic anisotropic curve-shortening flow, where the normal velocity at a point of its boundary is given by the local curvature times an explicit function of the local slope. The behavior should be similar at finite temperature T < Tc, with a different temperature-dependent anisotropy function.We prove this conjecture (at zero temperature) when D is convex. Existence and regularity of the solution of the deterministic curve-shortening flow is not obvious a priori and is part of our result. To our knowledge, this is the first proof of mean curvature-type droplet shrinking for a model with genuine microscopic dynamics.2000 Mathematics Subject Classification: 60K35, 82C20

In this paper we study the property of asymptotic direction for random walks in random i.i.d. environments (RWRE). We prove that if the set of directions where the walk is transient is non empty and open, the walk admits an asymptotic direction. The main tool to obtain this result is the construction of a renewal structure with cones. We also prove that RWRE admits at most two opposite asymptotic directions.Comment: revised versio