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We introduce a family of multi-dimensional Askey-Wilson signed measures. Weoffer an explicit description of the stationary measure of open asymmetricsimple exclusion process (ASEP) in the full phase diagram, in terms ofintegrations with respect to these Askey-Wilson signed measures. Using ourdescription, we provide a rigorous derivation of the density profile and limitfluctuations of open ASEP in the entire shock region, including the high andlow density phases as well as the coexistence line. This in particular confirmsthe existing physics postulations of the density profile.

Askey-Wilson signed measures and open ASEP in the shock region