Amine Asselah

From logarithmic to subdiffusive polynomial fluctuations for internal DLA and related growth models

We consider a cluster growth model on Zd, 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≥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 Blachere in our Appendix) on the expected time spent by a random walk inside an annulus.

Quenched Large Deviations for Diffusions in a Random Gaussian Shear Flow Drift

We prove a full large deviations principle for the one-dimensional laws of the diffusion process with random drift , where V is a centered Gaussian shear flow random field independent of the Brownian W. The large deviations principle is an annealed one, that is integrated over the randomnesses of V and W.

Fleming–Viot selects the minimal quasi-stationary distribution: The Galton–Watson case

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.

Sublogarithmic fluctuations for internal DLA

We consider internal diffusion limited aggregation in dimension larger than or equal to two. This is a random cluster growth model, where random walks start at the origin of the d-dimensional lattice, one at a time, and stop moving when reaching a site that is not occupied by previous walks. It is known that the asymptotic shape of the cluster is a sphere. When the dimension is two or more, we have shown in a previous paper that the inner (resp., outer) fluctuations of its radius is at most of order log(radius) [resp., log2(radius)]. Using the same approach, we improve the upper bound on the inner fluctuation to log(radius)−−−−−−−−−√ when d is larger than or equal to three. The inner fluctuation is then used to obtain a similar upper bound on the outer fluctuation.

First occurrence time of a large density fluctuation for a system of independent random walks

Abstract We obtain sharp asymptotics for the first time a “macroscopic” density fluctuation occurs in a system of independent simple symmetric random walks on Zd. Also, we show the convergence of the moments of the rescaled time by establishing tail estimates.

Fluctuations for internal DLA on the comb

We study internal diffusion limited aggregation (DLA) on the two dimensional comb lattice. The comb lattice is a spanning tree of the euclidean lattice, and internal DLA is a random growth model, where simple random walks, starting one at a time at the origin of the comb, stop when reaching the first unoccupied site. An asymptotic shape is suggested by a lower bound of Huss and Sava. We show that fluctuations with respect to this shape are gaussian as in the one-dimensional lattice.

Annealed upper tails for the energy of a charged polymer

We consider a randomly charged polymer. Each monomer carries a random charge, and only charges on the same site interact pairwise. We study the lower tails of the energy, when averaged over both randomness, in dimension three or more. As a corollary, we obtain the correct temperature-scale for the Gibbs measure.

Regularity of quasi-stationary measures for simple exclusion in dimension d ≥ 5

We consider the symmetric simple exclusion process on Z d , for d > 5, and study the regularity of the quasi-stationary measures of the dynamics conditioned on not occupying the origin. For each p ∈]0, 1[, we establish uniqueness of the density of quasi-stationary measures in L 2 (dν ρ ), where vp is the stationary measure of density p. This, in turn, permits us to obtain sharp estimates for P νρ (r > t), where r is the first time the origin is occupied.

Moderate deviations for the range of a transient random walk: path concentration

We study downward deviations of the boundary of the range of a transient walk on the Euclidean lattice. We describe the optimal strategy adopted by the walk in order to shrink the boundary of its range. The technics we develop apply equally well to the range, and provide pathwise statements for the {\it Swiss cheese} picture of Bolthausen, van den Berg and den Hollander \cite{BBH}.

Hitting times for independent random walks on ℤd

We consider a system of asymmetric independent random walks on Z d , denoted by {η t . t e R}, stationary under the product Poisson measure V p of marginal density p > 0. We fix a pattern A, an increasing local event, and denote by τ the hitting time of A. By using a loss network representation of our system, at small density, we obtain a coupling between the laws of η t conditioned on (r > t} for all times t. When d ≥ 3, this provides bounds on the rate of convergence of the law of η t conditioned on (r > t} toward its limiting probability measure as t tends to infinity. We also treat the case where the initial measure is close to V ρ without being product.