Systems of one-dimensional random walks in a common random environment

2009/07/21 by Jonathon Peterson, Peterson, Jonathon
Mathematics · #60K35 #60K37 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:60K35 #msc:60K37

paper · pdf · doi:10.48550/arxiv.0907.3680

15 pages

arxiv created 2016/06/11 · arxiv updated 2016/06/14

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 v_P. We give upper bounds on the quenched probability that at least one of the random walks started in the interval [An, Bn] has traveled a distance of less than (v_P - ε)n. This leads to both a uniform law of large numbers and a hydrodynamic limit. 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.

Related

• Hydrodynamic limit for a system of independent, sub-ballistic random walks in a common random environment
• Quenched Limits for Transient, Ballistic, Sub-Gaussian One-Dimensional Random Walk in Random Environment
• Weak quenched limiting distributions for transient one-dimensional random walk in a random environment
• Simple Transient Random Walks in One-dimensional Random Environment: the Central Limit Theorem
• Limiting distributions and large deviations for random walks in random environments