Diffusive limit of a tagged particle in asymmetric simple exclusion processes. (English) Zbl 1029.60084

Sethuraman, Sunder; Varadhan, S. R. S.; Yau, Horng-Tzer

Commun. Pure Appl. Math. 53, No. 8, 972-1006 (2000).

The authors study the motion of a tagged particle as it moves through other particles that are viewed as indistinguishable among themselves, in particular the fluctuation behaviour. The case of simple exclusion process on \(\mathbb Z^{d}\) when the underlying random walk has a symmetric distribution was considered by R. Arratia [Ann. Probab. 11, 362-373 (1983; Zbl 0515.60097)] for \(d=1\) and nearest-neighbour walks and by C. Kipnis and S. R. S. Varadhan [Commun. Math. Phys. 104, 1-19 (1986; Zbl 0588.60058)] for all other distributions and dimensions. The case of random walks with zero mean was treated by S. R. S. Varadhan [Ann. Inst. Henri Poincaré, Probab. Stat. 31, 273-285 (1995; Zbl 0816.60093)]. When the mean is not zero and \(d=1\), the study was performed by C. Kipnis [Ann. Probab. 14, 397-408 (1986; Zbl 0601.60098)]. In the present paper the authors study the case of dimensions \(d\geq 3\), and invariance principles are proved under diffusive scaling by considering the environment process as seen by the tagged particle. The authors explore the associated martingales and their proofs rely on the fact that the symmetric random walks are transient in high dimensions.

Reviewer: Mihai Gradinaru (Nancy)

MSC:

60K35	Interacting random processes; statistical mechanics type models; percolation theory
60F17	Functional limit theorems; invariance principles

Keywords:

motion of tagged particle; asymmetric simple exclusion process; resolvent equation

Citations:

Zbl 0515.60097; Zbl 0588.60058; Zbl 0816.60093; Zbl 0601.60098

References:

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