## Central limit theorems for infinite series of queues and applications to simple exclusion.(English)Zbl 0601.60098

In the simple exclusion process particles located at different sites of $${\mathbb{Z}}^ d$$ independently attempt to perform a continuous time random walk with exponential mean one waiting time. A jump is suppressed, if the site, at which the particle chooses to jump, is already occupied. The behaviour of the motion of a tagged particle, initially at the origin with the other sites being independently occupied with probability $$\rho$$, for large times has been studied for several cases with different limit behaviour [R. Arratia, ibid. 11, 362-373 (1983; Zbl 0515.60097) and the author and S. R. S. Varadhan, Commun. Math. Phys. 104, 1-19 (1986; Zbl 0588.60058)].
In this paper a central limit theorem is proved for the properly rescaled one-dimensional asymmetric nearest neighbour exclusion process with density $$\rho <1$$. The proof is based on the relation between this process and an infinite series of queues by studying the number of empty sites between the particles. For certain functions of the latter process central limit theorems are proved by deriving positive resp. negative correlation for them.
Reviewer: M.Mürmann

### MSC:

 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60F05 Central limit and other weak theorems

### Citations:

Zbl 0515.60097; Zbl 0588.60058
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