Mathematics – Probability
Scientific paper
2010-04-13
Ensaios Matem{\'a}ticos, Volume 18 (2010), 1-81
Mathematics
Probability
70 pages. Material for a minicourse at the 13th Brazilian School of Probability and at the University of Helsinki in August 20
Scientific paper
This review article discusses limit distributions and variance bounds for particle current in several dynamical stochastic systems of particles on the one-dimensional integer lattice: independent particles, independent particles in a random environment, the random average process, the asymmetric simple exclusion process, and a class of totally asymmetric zero range processes. The first three models possess linear macroscopic flux functions and lie in the Edwards-Wilkinson universality class with scaling exponent 1/4 for current fluctuations. For these we prove Gaussian limits for the current process. The latter two systems belong to the Kardar-Parisi-Zhang class. For these we prove the scaling exponent 1/3 in the form of upper and lower variance bounds.
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