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Current fluctuations for the totally asymmetric simple exclusion process. (English) Zbl 1015.60093
Sidoravicius, Vladas (ed.), In and out of equilibrium. Probability with a physics flavor. Papers from the 4th Brazilian school of probability, Mambucaba, Brazil, August 14-19, 2000. Boston: Birkhäuser. Prog. Probab. 51, 185-204 (2002).
The totally asymmetric simple exclusion process studied in the paper is as follows. Each particle jumps independently with rate 1 to the right neighboring site on the lattice line, provided it is empty. Denote by $$N_t$$ the number of particles which have crossed the bond $$(0, 1)$$ up to time $$t$$. It is clear that the mean of $$N_t$$ equals $$t/4$$. The purpose of the study is to describe the fluctuation $$N_t -t/4$$. The main result says that the fluctuation is non-Gaussin as $$t^{1/3}$$. The key idea comes from the recent progress on the connection of random matrices and the Painlevé II Riemann-Hilbert problem. Some open problems are also presented.
For the entire collection see [Zbl 0996.00040].

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory