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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].

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