Correlation and fluctuation in Random Average Process on an infinite line with a driven tracer

TL;DR

This paper investigates the large-time behavior of a biased tracer in a random average process on an infinite line, deriving scaling forms, correlation functions, and exact variance expressions supported by simulations.

Contribution

It provides a detailed analytical study of the correlations and fluctuations in RAP with a driven tracer, including exact results for the variance in the totally asymmetric case.

Findings

Scaling forms for mean and fluctuations grow as √t

Differential equation for correlation function solved perturbatively

Exact variance expression for totally asymmetric tracer

Abstract

We study the effect of single biased tracer particle in a bath of other particles performing the random average process (RAP) on an infinite line. We focus on the large time behavior of the mean and the fluctuations of the positions of the particles and also the correlations among them. In the large time t limit these quantities have well-defined scaling forms and grow with time as $\sqrt{t}$. A differential equation for the scaling function associated with the correlation function is obtained and solved perturbatively around the solution for a symmetric tracer. Interestingly, when the tracer is totally asymmetric, further progress is enabled by the fact that the particles behind of the tracer do not affect the motion of the particles in front of it, which leads in particular to an exact expression for the variance of the position of the tracer. Finally, the variance and correlations of the gaps between successive particles are also studied. Numerical simulations support our analytical results.