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Central limit theorems for additive functionals of the simple exclusion process. (English) Zbl 1044.60017

Summary: Some invariance principles for additive functionals of simple exclusion with finite-range translation-invariant jump rates \(p(i,j)= p(j-i)\) in dimensions \(d\geq 1\) are established. A previous investigation concentrated on the case of \(p\) symmetric. The principal tools to take care of nonreversibility, when \(p\) is asymmetric, are invariance principles for associated random variables and a “local balance” estimate on the asymmetric generator of the process. As a by-product, we provide upper and lower bounds on some transition probabilities for mean-zero asymmetric second-class particles, which are not Markovian, that show they behave like their symmetric Markovian counterparts. Also some estimates with respect to second-class particles with drift are discussed. In addition, a dichotomy between the occupation time process limits in \(d=1\) and \(d\geq 2\) for symmetric exclusion is shown. In the former, the limit is fractional Brownian motion with parameter \(3/4\), and in the latter, the usual Brownian motion.

MSC:

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