2013/05/04 by Frank den Hollander, Harry Kesten, Hollander, Frank den +3
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #60F05 #60K35 #Diffusion and Search Dynamics #FOS: Mathematics #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.1305.0923
openalex publication_date 2013/05/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \citeKeSi 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.