Exact gap statistics for the random average process on a ring with a tracer

TL;DR

This paper derives exact gap statistics for a symmetric random average process on a ring with a tracer particle, revealing universal distributions and correlations, and extends results to driven tracers with current-carrying steady states.

Contribution

It provides the first exact analytical results for gap distributions and correlations in RAP with a tracer, including driven cases, under specific conditions.

Findings

Universal factorized joint gap distribution for non-driven tracer

Exact single-site mass distribution in the thermodynamic limit

Precise two-point gap correlation functions

Abstract

We study statistics of the gaps in Random Average Process (RAP) on a ring with particles hopping symmetrically, except one tracer particle which could be driven. These particles hop either to the left or to the right by a random fraction $\eta$ of the space available till next particle in the respective directions. The random fraction $\eta \in [0,~1)$ is chosen from a distribution $R(\eta)$. For non-driven tracer, when $R(\eta)$ satisfies a necessary and sufficient condition, the stationary joint distribution of the gaps between successive particles takes an universal form that is factorized except for a global constraint. Some interesting explicit forms of $R(\eta)$ are found which satisfy this condition. In case of driven tracer, the system reaches a current-carrying steady state where such factorization does not hold. Analytical progress has been made in the thermodynamic limit, where we computed the single site mass distribution inside the bulk. We have also computed the two point gap-gap correlation exactly in that limit. Numerical simulations support our analytical results.