Current fluctuations for stochastic particle systems with drift in one spatial dimension

This review article discusses limit distributions and variance bounds for particle current in several dynamical stochastic systems of par- ticles on the one-dimensional integer lattice: independent particles, inde- pendent particles in a random environment, the random average process, the asymmetric simple exclusion process, and a class of totally asymmet- ric 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 Gaus- sian 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.