Abstract
We consider a system of independent one-dimensional random walks in a common random environment under the condition that the random walks are transient with positive speed. We give upper bounds on the quenched probability that at least one of the random walks started in an interval has experience a large deviation slowdown. This leads to both a uniform law of large numbers and a hydrodynamic limit for the system of random walks. We also identify a family of distributions on the configuration of particles (parameterized by particle density) which are stationary under the (quenched) dynamics of the random walks and show that these are the limiting distributions for the system when started from a certain natural collection of distributions.
Highlights
Introduction and Statement of the MainResultsThe object of study in this paper is a system of independent one-dimensional random walks in a common random environment
We modify the standard notion of random walks in random environment (RWRE) to allow for infinitely many particles
We let {X·x,i}x∈Z,i∈Z+ be an independent collection of Markov chains with law Pω defined by ωy z = y+1
Summary
The object of study in this paper is a system of independent one-dimensional random walks in a common random environment. The key to proving Theorem 1.1 is instead the following uniform analog of the quenched sub-exponential slowdown probabilities given in [5]. For slowdowns under the quenched measure, an environment is fixed and (with high probability) contains only smaller traps - making it harder to slow down the random walk. We will allow the initial distribution to depend on the environment ω (in a measurable way), but given ω we will require that the initial configuration is a product measure To make this precise, let Υ be the space of probability distributions on the non-negative integers. Let Pω,νω denote the quenched distribution of the system of random walks with initial distribution given by νω, and let Pν(·) = Ω Pω,νω (·)P (dω) be the corresponding averaged distribution.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
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 callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
