Exact Tagged Particle Correlations in the Random Average Process

Abstract

We study analytically the correlations between the positions of tagged particles in the random average process, an interacting particle system in one dimension. We show that in the steady state the mean squared auto-fluctuation of a tracer particle grows subdiffusively as $sigma^2(t) ~ t^{1/2}$ for large time t in the absence of external bias, but grows diffusively $sigma^2(t) ~ t$ in the presence of a nonzero bias. The prefactors of the subdiffusive and diffusive growths as well as the universal scaling function describing the crossover between them are computed exactly. We also compute $sigma_r^2(t)$, the mean squared fluctuation in the position difference of two tagged particles separated by a fixed tag shift r in the steady state and show that the external bias has a dramatic effect in the time dependence of $sigma_r^2(t)$. For fixed r, $sigma_r^2(t)$ increases monotonically with t in absence of bias but has a non-monotonic dependence on t in presence of bias. Similarities and differences with the simple exclusion process are also discussed.