Limit fluctuations for density of asymmetric simple exclusion processes with open boundaries

TL;DR

This paper characterizes the scaling limits of density fluctuations in the asymmetric simple exclusion process with open boundaries across different phases, revealing complex limiting processes involving Brownian motion, excursions, and meanders.

Contribution

It provides a complete description of the fluctuation limits in all phases, extending previous results and introducing new boundary fluctuation behaviors.

Findings

Maximal current phase: fluctuations are sum of Brownian motion and Brownian excursion.

Low/high density phases: fluctuations are Brownian motion.

Boundary of maximal current phase: fluctuations are sum of Brownian motion and Brownian meander.

Abstract

We investigate the fluctuations of cumulative density of particles in the asymmetric simple exclusion process with respect to the stationary distribution (also known as the steady state), as a stochastic process indexed by $[0,1]$. In three phases of the model and their boundaries within the fan region, we establish a complete picture of the scaling limits of the fluctuations of the density as the number of sites goes to infinity. In the maximal current phase, the limit fluctuation is the sum of two independent processes, a Brownian motion and a Brownian excursion. This extends an earlier result by Derrida et al. (2004) for totally asymmetric simple exclusion process in the same phase. In the low/high density phases, the limit fluctuations are Brownian motion. Most interestingly, at the boundary of the maximal current phase, the limit fluctuation is the sum of two independent processes, a Brownian motion and a Brownian meander (or a time-reversal of the latter, depending on the side of the boundary). Our proofs rely on a representation of the joint generating function of the asymmetric simple exclusion process with respect to the stationary distribution in terms of joint moments of a Markov processes, which is constructed from orthogonality measures of the Askey--Wilson polynomials.