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Random walk on random walks. (English) Zbl 1328.60226
Summary: 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 \(\rho \in (0,\infty)\). At each step the random walk performs a nearest-neighbour jump, moving to the right with probability \(p_{\circ}\) when it is on a vacant site and probability \(p_{\bullet}\) when it is on an occupied site. Assuming that \(p_\circ \in (0,1)\) and \(p_\bullet \neq \frac 12\), 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 \(\rho\) is large enough. The proof is based on the construction of a renewal structure together with a multiscale renormalisation argument.

MSC:
60K37 Processes in random environments
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60G50 Sums of independent random variables; random walks
60F15 Strong limit theorems
60F17 Functional limit theorems; invariance principles
60F05 Central limit and other weak theorems
60F10 Large deviations
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
82C22 Interacting particle systems in time-dependent statistical mechanics
82C44 Dynamics of disordered systems (random Ising systems, etc.) in time-dependent statistical mechanics
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