Random walk on random walks

Abstract

In this paper we study a random walk in a one-dimensional dynamic random environment consisting of a collection of independent particles performing simple symmetric random walks in a Poisson equilibrium with density ρ ∈ (0, ∞). At each step the random walk performs a nearest-neighbour jump, moving to the right with probability p • when it is on a vacant site and probability p • when it is on an occupied site. Assuming that p • ∈ (0, 1) and p • = 1 2 , we show that the position of the random walk satisfies a strong law of large numbers, a functional central limit theorem and a large deviation bound, provided ρ is large enough. The proof is based on the construction of a renewal structure together with a multiscale renormalisation argument.MSC 2010. Primary 60F15, 60K35, 60K37; Secondary 82B41, 82C22, 82C44.

Response to a small external force and fluctuations of a passive particle in a one-dimensional diffusive environment

“…Lattice models. Lattice versions of the evolution equation (1) have been considered both in the probabilist community [16][17][18][19][20] and in numerical studies, e.g. [12,13,15].…”

“…Remark on transient behavior. The expressions (17)(18)(19) and (21)(22)(23) only hold in the limit t → ∞ at fixed values of all other parameters, and may have to be modified on some transient time scales. E.g.…”

“…Moreover, it is a rather natural set-up to consider since diffusive fluctuations occur in all typical extended systems that satisfy local equilibrium and have extensive conserved quantities. However, predicting the long time behavior of the passive particle turns out to be very puzzling in d = 1 [14][15][16][17][18][19][20][21] (in contrast to a lot of progress made for divergent free fields in d ≥ 2 [22][23][24][25]). Indeed, since time correlations decay only as t −d/2 , one expects memory effects to play a dominant role in d = 1, but it is hard to decide what their influence actually is.…”

Random walk on random walks: higher dimensions

“…But arguably, 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). Each of these works imposes additional conditions and explores very specific properties of the environment in question.…”

“…In the present paper, we consider as in [16] dynamic random environments given by systems of independent simple symmetric random walks. As mentioned above, asymptotic results for this model are challenging since the random environment is conservative and has slow and non-uniform mixing.…”