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

Consider a continuous-time Markov process with transition rates matrix Q in the state space ∪ {0}. In the associated Fleming-Viot process N particles evolve independently in with transition rates matrix Q until one of them attempts to jump to state 0. At this moment the particle jumps to one of the positions of the other particles, chosen uniformly at random. When is finite, we show that the empirical distribution of the particles at a fixed time converges as N → ∞ to the distribution of a single particle at the same time conditioned on not touching {0}. Furthermore, the empirical profile of the unique invariant measure for the Fleming-Viot process with N particles converges as N → ∞ to the unique quasistationary distribution of the one-particle motion. A key element of the approach is to show that the two-particle correlations are of order 1/N .

We study the capacity of the range of a transient simple random walk on Z d . Our main result is a central limit theorem for the capacity of the range for d ≥ 6. We present a few open questions in lower dimensions.

Consider N particles moving independently, each one according to a subcritical continuous-time Galton-Watson process unless it hits 0, at which time it jumps instantaneously to the position of one of the other particles chosen uniformly at random. The resulting dynamics is called Fleming-Viot process. We show that for each N there exists a unique invariant measure for the Fleming-Viot process, and that its stationary empirical distribution converges, as N goes to infinity, to the minimal quasi-stationary distribution of the Galton-Watson process conditioned on non-extinction.

Let {S k , k ≥ 0} be a symmetric random walk on Z d , and {η(x), x ∈ Z d } an independent random field of centered i.i.d. random variables with tail decayWe consider a random walk in random scenery, that is X n = η(S 0 ) + · · · + η(S n ). We present asymptotics for the probability, over both randomness, that {X n > n β } for β > 1/2 and α > 1. To obtain such asymptotics, we establish large deviations estimates for the self-intersection local times process x l 2 n (x), where l n (x) is the number of visits of site x up to time n.