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

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.

31 pages; minor revision

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• Nonequilibrium Steady States of Matrix Product Form: A Solver's Guide
• Asymmetric Simple Exclusion Process with open boundaries and Quadratic Harnesses
• Dual representations of Laplace transforms of Brownian excursion and generalized meanders
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Cited by in corpus (9)

• From the asymmetric simple exclusion processes to the stationary measures of the KPZ fixed point on an interval
• KPZ fluctuations in finite volume
• Askey-Wilson signed measures and open ASEP in the shock region
• Steady state of the KPZ equation on an interval and Liouville quantum mechanics
• Fluctuations of random Motzkin paths II
• Mixing times and cutoff for the TASEP in the high and low density phase
• Stationary measure for six-vertex model on a strip
• Limit fluctuations of stationary measure of totally asymmetric simple exclusion process with open boundaries on the coexistence line
• Limits of open ASEP stationary measures near a boundary