Greedy clearing of persistent Poissonian dust

Given a Poisson point process on R, assign either one or two marks to each point of this process, independently of the others. We study the motion of a particle that jumps deterministically from its current location to the nearest point of the Poisson point process which still contains at least one mark, and removes one mark per each visit. A point of the Poisson point process which is left with no marks is removed from the system. We prove that the presence of any positive density of double marks leads to the eventual removal of every Poissonian point.Fil: Trivellato Rolla, Leonardo. Conselho Nacional de Desenvolvimiento Cientf y Tec. Associacao Instituto Nacional de Matemática Pura E Aplicada; Brasil. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santalo". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santalo"; ArgentinaFil: Sidoravicius, V.. Conselho Nacional de Desenvolvimiento Cientf y Tec. Associacao Instituto Nacional de Matemática Pura E Aplicada; BrasilFil: Tournier, Laurent. Universite de Paris 13-Nord; Franci

Law of large numbers for non-elliptic random walks in dynamic random environments

Article / Letter to editorMathematisch Instituu

Non-equilibrium Phase Transitions: Activated Random Walks at Criticality

In this paper we present rigorous results on the critical behavior of the Activated Random Walk model. We conjecture that on a general class of graphs, including Z d , and under general initial conditions, the system at the critical point does not reach an absorbing state. We prove this for the case where the sleep rate λ is infinite. Moreover, for the one-dimensional asymmetric system, we identify the scaling limit of the flow through the origin at criticality. The case λ < +∞ remains largely open, with the exception of the one-dimensional totally-asymmetric case, for which it is known that there is no fixation at criticality.Fil: Cabezas, M.. Conselho Nacional de Desenvolvimiento Cientf y Tec. Associacao Instituto Nacional de Matemática Pura E Aplicada; BrasilFil: Trivellato Rolla, Leonardo. Conselho Nacional de Desenvolvimiento Cientf y Tec. Associacao Instituto Nacional de Matemática Pura E Aplicada; Brasil. Universidad de Buenos Aires; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Sidoravicius, V.. Conselho Nacional de Desenvolvimiento Cientf y Tec. Associacao Instituto Nacional de Matemática Pura E Aplicada; Brasi

Random walk on random walks

In this paper we study a random walk in a one-dimensional dynamic random environment consisting of a collection of independent particles performing simple symmetric random walks in a Poisson equilibrium with density ¿¿(0,8). At each step the random walk performs a nearest-neighbour jump, moving to the right with probability p° when it is on a vacant site and probability p· when it is on an occupied site. Assuming that p°¿(0,1) and p·¿12, we show that the position of the random walk satisfies a strong law of large numbers, a functional central limit theorem and a large deviation bound, provided ¿ is large enough. The proof is based on the construction of a renewal structure together with a multiscale renormalisation argument

Random walk on random walks

In this paper we study a random walk in a one-dimensional dynamic random environment consisting of a collection of independent particles performing simple symmetric random walks in a Poisson equilibrium with density ¿¿(0,8). At each step the random walk performs a nearest-neighbour jump, moving to the right with probability p° when it is on a vacant site and probability p· when it is on an occupied site. Assuming that p°¿(0,1) and p·¿12, we show that the position of the random walk satisfies a strong law of large numbers, a functional central limit theorem and a large deviation bound, provided ¿ is large enough. The proof is based on the construction of a renewal structure together with a multiscale renormalisation argument

Stochastic perturbations of convex billiards

We consider a strictly convex billiard table with C 2 boundary, with the dynamics subjected to random perturbations. Each time the billiard ball hits the boundary its reflection angle has a random perturbation. The perturbation distribution corresponds to the physical situation where either the scale of the surface irregularities is smaller than but comparable to the diameter of the reflected object, or the billiard ball is not perfectly rigid. We prove that for a large class of such perturbations the resulting Markov chain is uniformly ergodic, although this is not true in general.Fil: Markarian, R.. Universidad de la Republica. Facultad de Ingeniería; UruguayFil: Trivellato Rolla, Leonardo. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santalo". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santalo"; ArgentinaFil: Sidoravicius, V.. Jardim Botânico; BrasilFil: Tal, F. A.. Universidade de Sao Paulo; BrasilFil: Vares, M. E.. Universidade Federal do Rio de Janeiro; Brasi

Oriented Percolation in One-Dimensional 1/|x-y|^2 Percolation Models

We consider independent edge percolation models on Z, with edge occupation probabilities p_ = p if |x-y| = 1, 1 - exp{- beta / |x-y|^2} otherwise. We prove that oriented percolation occurs when beta > 1 provided p is chosen sufficiently close to 1, answering a question posed in [Commun. Math. Phys. 104, 547 (1986)]. The proof is based on multi-scale analysis.Comment: 19 pages, 2 figures. See also Commentary on J. Stat. Phys. 150, 804-805 (2013), DOI 10.1007/s10955-013-0702-