Limits of open ASEP stationary measures near a boundary

Consider the stationary measure of open asymmetric simple exclusion process (ASEP) on the lattice $\{1,\dots ,n\}$. 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 provide bounds on the convergence rates in total variation distance. 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.