Limits of open ASEP stationary measures near a boundary

Consider the stationary measure of open asymmetric simple exclusion process (ASEP) on the lattice \1,,n\. Taking n to infinity while fixing the jump rates, this measure converges to a measure on the semi-infinite lattice. In the high and low density phases, we characterize the limiting measure and also show that this convergence occurs in total variation distance on a sublattice of scale n/log n. Our approach involves bounding the total variation distance using generating functions, which are further estimated through a subtle analysis of the atom masses of Askey-Wilson signed measures.