Annealed upper tails for the energy of a polymer

We study the upper tails for the energy of a randomly charged symmetric and transient random walk. We assume that only charges on the same site interact pairwise. We consider annealed estimates, that is when we average over both randomness, in dimension three or more. We obtain a large deviation principle, and an explicit rate function for a large class of charge distributions.Comment: 36 pages, focus on upper tails; the lower tails estimates make another pape

On large intersection and self-intersection local times in dimension five or more

We show a remarkable similarity between strategies to realize a large intersection or self-intersection local times in dimension five or more. This leads to the same rate functional for large deviation principles for the two objects obtained respectively by Chen and Morters, and by the present author. We also present a new estimate for the distribution of high level sets for a random walk, with application to the geometry of the intersection set of two high level sets of the local times of two independent random walks.Comment: 16 page

On the Dirichlet problem for asymmetric zero-range process on increasing domains

We characterize the principal eigenvalue of the generator of the asymmetric zero-range process in dimensions d>2, with Dirichlet boundary on special domains. We obtain a Donsker-Varadhan variational representation for the principal eigenvalue, and show that the corresponding eigenfunction is unique in a natural class of functions. This allows us to obtain asymptotic hitting time estimates.Comment: 33 pages http://www.cmi.univ-mrs.fr/~assela

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 $\log(\mathrm{radius})$ [resp., $\log^2(\mathrm{radius})$]. Using the same approach, we improve the upper bound on the inner fluctuation to $\sqrt{\log(\mathrm{radius})}$ 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.Comment: Published in at http://dx.doi.org/10.1214/11-AOP735 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

On the nature of the Swiss cheese in dimension 3

We study scenarii linked with the Swiss cheese picture in dimension three obtained when two random walks are forced to meet often, or when one random walk is forced to squeeze its range. In the case of two random walks, we show that they most likely meet in a region of optimal density. In the case of one random walk, we show that a small range is reached by a strategy uniform in time. Both results rely on an original inequality estimating the cost of visiting sparse sites, and in the case of one random walk on the precise Large Deviation Principle of van den Berg, Bolthausen and den Hollander, including their sharp estimates of the rate functions in the neighborhood of the originComment: 11 page

On outer fluctuations for internal DLA

We had established inner and outer fluctuation for the internal DLA cluster when all walks are launched from the origin. In obtaining the outer fluctuation, we had used a deep lemma of Jerison, Levine and Sheffield, which estimate roughly the possibility of fingering, and had provided a simple proof using an interesting estimate for crossing probability for a simple random walk. The application of the crossing probability to the fingering for the internal DLA cluster contains a flaw discovered recently, that we correct in this note. We take the opportunity to make a self-contained exposition.Comment: 10 page

Boundary of the Range of Transient Random Walk

We study the boundary of the range of simple random walk on $\mathbb{Z}^d$ in the transient regime $d\ge 3$. We show that volumes of the range and its boundary differ mainly by a martingale. As a consequence, we obtain a bound on the variance of order $n\log n$ in dimension three. We also establish a central limit theorem in dimension four and larger.Comment: 23 pages. Revised Versio