Sharp approximations for the stationary measures of the open ASEP II: The coexistence line

Abstract

We study the stationary measure of the open asymmetric simple exclusion process (ASEP) on the coexistence line. Our starting point is Bryc's two-layer Gibbs representation, together with the probabilistic interpretation, established in the first paper in this series, of the associated Bryc polynomial in terms of an inhomogeneous birth--death chain. Our main result gives a sharp approximation of the stationary measure. Specifically, we identify the shock location as an explicit observable of the particle configuration and obtain optimal bounds on the total variation distance between its law and the uniform measure on the lattice. Conditional on the shock location, we approximate, in total variation distance, the joint law of the occupation variables away from the shock by Bernoulli product measures with different densities to its left and right. These approximations lead to further applications, including fluctuation limits, moderate deviation bounds for the particle density, and limits for correlation functions under the stationary measure.