Abstract
We study the evolution of a random walker on a conservative dynamic random environment composed of independent particles performing simple symmetric random walks, generalizing results of [16] to higher dimensions and more general transition kernels without the assumption of uniform ellipticity or nearest-neighbour jumps. Specifically, we obtain a strong law of large numbers, a functional central limit theorem and large deviation estimates for the position of the random walker under the annealed law in a high density regime. The main obstacle is the intrinsic lack of monotonicity in higher-dimensional, non-nearest neighbour settings. Here we develop more general renormalization and renewal schemes that allow us to overcome this issue. As a second application of our methods, we provide an alternative proof of the ballistic behaviour of the front of (the discrete-time version of) the infection model introduced in [23].
Highlights
Random walks on random environments are models for the movement of a tracer particle in a disordered medium, and have been the subject of intense research for over 40 years
The dynamic version of the model, i.e., when the random environment is allowed to evolve in time, has been studied for over three decades
A method to overcome this difficulty in ballistic situations was developed in [16] for the high density regime in one dimension, see [21] for a similar approach when the random environment is given by a one-dimensional simple symmetric exclusion process
Summary
Random walks on random environments are models for the movement of a tracer particle in a disordered medium, and have been the subject of intense research for over 40 years. Asymptotic results for RWDRE under general conditions were derived e.g. in [5, 6, 12, 15, 19, 29, 30], often requiring uniform mixing conditions on the random environment (implying e.g. that the conditional distribution of the environment at the origin given the initial state uniformly approaches a fixed law for large times). Some of the most challenging random environments are given by conservative particle systems, due to their poor mixing properties Such cases have been considered in [4, 7, 8, 21, 32] (simple symmetric exclusion), and in [16, 17] (independent random walks). Some tools developed in the present paper will be used in the accompanying article [13]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
