Walks In Dynamic Random Environments Research Articles

Collisions of random walks in dynamic random environments

We study dynamic random conductance models on Z2 in which the environment evolves as a reversible Markov process that is stationary under space-time shifts. We prove under a second moment assumption that two conditionally independent random walks in the same environment collide infinitely often almost surely. These results apply in particular to random walks on dynamical percolation.

Regular and Singular Continuous Time Random Walk in Dynamic Random Environment

Regular and Singular Continuous Time Random Walk in Dynamic Random Environment

Analysis of random walks in dynamic random environments via [formula omitted]-perturbations

We consider random walks in dynamic random environments given by Markovian dynamics on Zd. We assume that the environment has a stationary distribution μ and satisfies the Poincaré inequality w.r.t. μ. The random walk is a perturbation of another random walk (called “unperturbed”). We assume that also the environment viewed from the unperturbed random walk has stationary distribution μ. Both perturbed and unperturbed random walks can depend heavily on the environment and are not assumed to be finite-range. We derive a law of large numbers, an averaged invariance principle for the position of the walker and a series expansion for the asymptotic speed. We also provide a condition for non-degeneracy of the diffusion, and describe in some details equilibrium and convergence properties of the environment seen by the walker. All these results are based on a more general perturbative analysis of operators that we derive in the context of L2- bounded perturbations of Markov processes by means of the so-called Dyson–Phillips expansion.

Zero-one law for directional transience of one-dimensional random walks in dynamic random environments

We prove the trichotomy between transience to the right, transience to the left and recurrence of one-dimensional nearest-neighbour random walks in dynamic random environments under fairly general assumptions, namely: stationarity under space-time translations, ergodicity under spatial translations, and a mild ellipticity condition. In particular, the result applies to general uniformly elliptic models and also to a large class of non-uniformly elliptic cases that are i.i.d. in space and Markovian in time. An immediate consequence is the recurrence of models that are symmetric with respect to reflection through the origin.

Random walks in dynamic random environments and ancestry under local population regulation

We consider random walks in dynamic random environments, with an environment generated by the time-reversal of a Markov process from the oriented percolation universality class. If the influence of the random medium on the walk is small in space-time regions where the medium is typical, we obtain a law of large numbers and an averaged central limit theorem for the walk via a regeneration construction under suitable coarse-graining. Such random walks occur naturally as spatial embeddings of ancestral lineages in spatial population models with local regulation. We verify that our assumptions hold for logistic branching random walks when the population density is sufficiently high.

Absolute continuity and weak uniform mixing of random walk in dynamic random environment

We prove results for random walks in dynamic random environments which do not require the strong uniform mixing assumptions present in the literature. We focus on the “environment seen from the walker-process and in particular its invariant law. Under general conditions it exists and is mutually absolutely continuous to the environment law. With stronger assumptions we obtain for example uniform control on the density or a quenched CLT. The general conditions are made more explicit by looking at hidden Markov models or Markov chains as environment and by providing simple examples.

Random Walk in Dynamic Random Environment with Long-Range Space Correlations

Random Walk in Dynamic Random Environment with Long-Range Space Correlations

Random walks on weighted, oriented percolation clusters

Abstract. We consider a weighted random walk on the backbone of an oriented percolation cluster. We determine necessary conditions on the weights for Brownian scaling limits under the annealed and the quenched law. This model is a random walk in dynamic random environment (RWDRE), where the environment is mixing, non-Markovian and not elliptic. We provide a generalization of results obtained previously by Birkner et al. (2013).

Random walk in a high density dynamic random environment

The goal of this note is to prove a law of large numbers for the empirical speed of a green particle that performs a random walk on top of a field of red particles which themselves perform independent simple random walks on Zd, d≥1. The red particles jump at rate 1 and are in a Poisson equilibrium with density μ. The green particle also jumps at rate 1, but uses different transition kernels p′ and p″ depending on whether it sees a red particle or not. It is shown that, in the limit as μ→∞, the speed of the green particle tends to the average jump under p′. This result is far from surprising, but it is non-trivial to prove. The proof that is given in this note is based on techniques that were developed in Kesten and Sidoravicius (2005) to deal with spread-of-infection models. The main difficulty is that, due to particle conservation, space–time correlations in the field of red particles decay slowly. This places the problem in a class of random walks in dynamic random environments for which scaling laws are hard to obtain.

Random walks in dynamic random environments: A transference principle

We study a general class of random walks driven by a uniquely ergodic Markovian environment. Under a coupling condition on the environment we obtain strong ergodicity properties for the environment as seen from the position of the walker, that is, the environment process. We can transfer the rate of mixing in time of the environment to the rate of mixing of the environment process with a loss of at most polynomial order. Therefore the method is applicable to environments with sufficiently fast polynomial mixing. We obtain unique ergodicity of the environment process. Moreover, the unique invariant measure of the environment process depends continuously on the jump rates of the walker. As a consequence we obtain the law of large numbers and a central limit theorem with nondegenerate variance for the position of the walk.

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

We prove a law of large numbers for a class of Zd-valued random walks in dynamic random environments, including non-elliptic examples. We assume for the random environment a mixing property called conditional cone-mixing and that the random walk tends to stay inside wide enough space–time cones. The proof is based on a generalization of a regeneration scheme developed by Comets and Zeitouni (2004) [5] for static random environments and adapted by Avena et al. (2011) [2] to dynamic random environments. A number of one-dimensional examples are given. In some cases, the sign of the speed can be determined.

Continuity and Anomalous Fluctuations in Random Walks in Dynamic Random Environments: Numerics, Phase Diagrams and Conjectures

We perform simulations for one dimensional continuous-time random walks in two dynamic random environments with fast (independent spin-flips) and slow (simple symmetric exclusion) decay of space-time correlations, respectively. We focus on the asymptotic speeds and the scaling limits of such random walks. We observe different behaviors depending on the dynamics of the underlying random environment and the ratio between the jump rate of the random walk and the one of the environment. We compare our data with well known results for static random environment. We observe that the non-diffusive regime known so far only for the static case can occur in the dynamic setup too. Such anomalous fluctuations give rise to a new phase diagram. Further we discuss possible consequences for more general static and dynamic random environments.

Law of Large Numbers for a Class of Random Walks in Dynamic Random Environments

In this paper we consider a class of one-dimensional interacting particle systems in equilibrium, constituting a dynamic random environment, together with a nearest-neighbor random walk that on occupied/vacant sites has a local drift to the right/left. We adapt a regeneration-time argument originally developed by Comets and Zeitouni for static random environments to prove that, under a space-time mixing property for the dynamic random environment called cone-mixing, the random walk has an a.s. constant global speed. In addition, we show that if the dynamic random environment is exponentially mixing in space-time and the local drifts are small, then the global speed can be written as a power series in the size of the local drifts. From the first term in this series the sign of the global speed can be read off. The results can be easily extended to higher dimensions.