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Random spatial growth with paralyzing obstacles

J. van den Berg, Y. Peres, V. Sidoravicius, M. E. Vares (2008)

Annales de l'I.H.P. Probabilités et statistiques

We study models of spatial growth processes where initially there are sources of growth (indicated by the colour green) and sources of a growth-stopping (paralyzing) substance (indicated by red). The green sources expand and may merge with others (there is no ‘inter-green’ competition). The red substance remains passive as long as it is isolated. However, when a green cluster comes in touch with the red substance, it is immediately invaded by the latter, stops growing and starts to act as a red...

Random walk in random environment with asymptotically zero perturbation

M.V. Menshikov, Andrew R. Wade (2006)

Journal of the European Mathematical Society

We give criteria for ergodicity, transience and null-recurrence for the random walk in random environment on ℤ + = { 0 , 1 , 2 , ⋯ } , with reflection at the origin, where the random environment is subject to a vanishing perturbation. Our results complement existing criteria for random walks in random environments and for Markov chains with asymptotically zero drift, and are significantly different from the previously studied cases. Our method is based on a martingale technique—the method of Lyapunov functions.

Random Walks in Attractive Potentials: The Case of Critical Drifts

Dmitry Ioffe, Yvan Velenik (2010)

Actes des rencontres du CIRM

We consider random walks in attractive potentials - sub-additive functions of their local times. An application of a drift to such random walks leads to a phase transition: If the drift is small than the walk is still sub-ballistic, whereas the walk is ballistic if the drift is strong enough. The set of sub-critical drifts is convex with non-empty interior and can be described in terms of Lyapunov exponents (Sznitman, Zerner ). Recently it was shown that super-critical drifts lead to a limiting...

Regularity of the effective diffusivity of random diffusion with respect to anisotropy coefficient

M. Cudna, T. Komorowski (2008)

Studia Mathematica

We show that the effective diffusivity of a random diffusion with a drift is a continuous function of the drift coefficient. In fact, in the case of a homogeneous and isotropic random environment the function is C ∞ smooth outside the origin. We provide a one-dimensional example which shows that the diffusivity coefficient need not be differentiable at 0.

Reinforced walk on graphs and neural networks

Józef Myjak, Ryszard Rudnicki (2008)

Studia Mathematica

A directed-edge-reinforced random walk on graphs is considered. Criteria for the walk to end up in a limit cycle are given. Asymptotic stability of some neural networks is shown.

Reversed Dirichlet environment and directional transience of random walks in Dirichlet environment

Christophe Sabot, Laurent Tournier (2011)

Annales de l'I.H.P. Probabilités et statistiques

We consider random walks in a random environment given by i.i.d. Dirichlet distributions at each vertex of ℤd or, equivalently, oriented edge reinforced random walks on ℤd. The parameters of the distribution are a 2d-uplet of positive real numbers indexed by the unit vectors of ℤd. We prove that, as soon as these weights are nonsymmetric, the random walk is transient in a direction (i.e., it satisfies Xn ⋅ ℓ →n +∞ for some ℓ) with positive probability. In dimension 2, this result is strenghened...

Scaling limit of the random walk among random traps on ℤd

Jean-Christophe Mourrat (2011)

Annales de l'I.H.P. Probabilités et statistiques

Attributing a positive value τx to each x∈ℤd, we investigate a nearest-neighbour random walk which is reversible for the measure with weights (τx), often known as “Bouchaud’s trap model.” We assume that these weights are independent, identically distributed and non-integrable random variables (with polynomial tail), and that d≥5. We obtain the quenched subdiffusive scaling limit of the model, the limit being the fractional kinetics process. We begin our proof by expressing the random walk as the...

Scaling of a random walk on a supercritical contact process

F. den Hollander, R. S. dos Santos (2014)

Annales de l'I.H.P. Probabilités et statistiques

We prove a strong law of large numbers for a one-dimensional random walk in a dynamic random environment given by a supercritical contact process in equilibrium. The proof uses a coupling argument based on the observation that the random walk eventually gets trapped inside the union of space–time cones contained in the infection clusters generated by single infections. In the case where the local drifts of the random walk are smaller than the speed at which infection clusters grow, the random walk...

Slowdown estimates for ballistic random walk in random environment

Noam Berger (2012)

Journal of the European Mathematical Society

We consider models of random walk in uniformly elliptic i.i.d. random environment in dimension greater than or equal to 4, satisfying a condition slightly weaker than the ballisticity condition ( T ' ) . We show that for every ϵ > 0 and n large enough, the annealed probability of linear slowdown is bounded from above by exp ( - ( log n ) d - ϵ ) . This bound almost matches the known lower bound of exp ( - C ( log n ) d ) , and significantly improves previously known upper bounds. As a corollary we provide almost sharp estimates for the quenched probability...

Small positive values for supercritical branching processes in random environment

Vincent Bansaye, Christian Böinghoff (2014)

Annales de l'I.H.P. Probabilités et statistiques

Branching Processes in Random Environment (BPREs) ( Z n : n ≥ 0 ) are the generalization of Galton–Watson processes where in each generation the reproduction law is picked randomly in an i.i.d. manner. In the supercritical case, the process survives with positive probability and then almost surely grows geometrically. This paper focuses on rare events when the process takes positive but small values for large times. We describe the asymptotic behavior of ℙ ( 1 ≤ Z n ≤ k | Z 0 = i ) , k , i ∈ ℕ as n → ∞ . More precisely, we characterize the exponential...

Superdiffusivity for brownian motion in a poissonian potential with long range correlation I: Lower bound on the volume exponent

Hubert Lacoin (2012)

Annales de l'I.H.P. Probabilités et statistiques

We study trajectories of d -dimensional Brownian Motion in Poissonian potential up to the hitting time of a distant hyper-plane. Our Poissonian potential V is constructed from a field of traps whose centers location is given by a Poisson Point Process and whose radii are IID distributed with a common distribution that has unbounded support; it has the particularity of having long-range correlation. We focus on the case where the law of the trap radii ν has power-law decay and prove that superdiffusivity...

Currently displaying 81 – 100 of 119