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We consider the asymmetric simple exclusion process (ASEP) on the positiveintegers with an open boundary condition. We show that, when starting devoid ofparticles and for a certain boundary condition, the height function at theorigin fluctuates asymptotically (in large time $\tau$) according to theTracy-Widom GOE distribution on the $\tau^{1/3}$ scale. This is the firstexample of KPZ asymptotics for a half-space system outside the class offree-fermionic/determinantal/Pfaffian models.  Our main tool in this analysis is a new class of probability measures onYoung diagrams that we call half-space Macdonald processes, as well as twosurprising relations. The first relates a special (Hall-Littlewood) case ofthese measures to the half-space stochastic six-vertex model (which furtherlimits to ASEP) using a Yang-Baxter graphical argument. The second relatescertain averages under these measures to their half-space (or Pfaffian) Schurprocess analogs via a refined Littlewood identity.

Stochastic six-vertex model in a half-quadrant and half-line open ASEP