Sharp approximations for the stationary measures of the open ASEP III: The fan region

We study the stationary measure of the open asymmetric simple exclusion process (ASEP) in the fan region with general boundary rates. The stationary height fluctuations in the maximal current phase and on its boundaries within the fan region are known to have limits involving Brownian excursions or meanders. We establish discrete counterparts of these limits: on intervals away from the boundaries, the joint law of the occupation variables is approximated in total variation distance by particle configurations constructed from lazy simple random walk excursions and meanders. In the high- and low-density phases of the fan region, we establish Bernoulli product approximations with errors that decay exponentially in the distance from the relevant boundary. Our methods combine refined combinatorial estimates for Motzkin paths and the matrix product ansatz in the totally asymmetric case with a Fourier-type decomposition (based on Askey--Wilson polynomials) of the change of measure for general asymmetry. In addition, in the maximal current phase and on its boundaries within the fan region, we prove moderate deviation estimates for particle counts on arbitrary sets of sites. These estimates rely on a covariance identity for an auxiliary path measure, through which total nonnegativity (of order two) of the matrices yields negative correlations.