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

Abstract. 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.