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

TL;DR

This review explores the fluctuation behavior of particle currents in various one-dimensional stochastic systems, highlighting Gaussian limits in some models and KPZ class scaling in others, with emphasis on variance bounds and universality classes.

Contribution

It provides a comprehensive comparison of current fluctuations across multiple models, establishing Gaussian limits for some and variance bounds for KPZ class systems.

Findings

Gaussian limits for models with linear flux functions

Variance bounds indicating 1/3 scaling in KPZ class models

Different universality classes exhibit distinct fluctuation behaviors

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.