We consider the asymmetric simple exclusion process (ASEP) on the positive integers with an open boundary condition. We show that, when starting devoid of particles and for a certain boundary condition, the height function at the origin fluctuates asymptotically (in large time ) according to the Tracy–Widom Gaussian orthogonal ensemble distribution on the -scale. This is the first example of Kardar–Parisi–Zhang asymptotics for a half-space system outside the class of free-fermionic/determinantal/Pfaffian models.
Our main tool in this analysis is a new class of probability measures on Young diagrams that we call half-space Macdonald processes, as well as two surprising relations. The first relates a special (Hall–Littlewood) case of these measures to the half-space stochastic six-vertex model (which further limits to the ASEP) using a Yang–Baxter graphical argument. The second relates certain averages under these measures to their half-space (or Pfaffian) Schur process analogues via a refined Littlewood identity.
"Stochastic six-vertex model in a half-quadrant and half-line open asymmetric simple exclusion process." Duke Math. J. 167 (13) 2457 - 2529, 15 September 2018. https://doi.org/10.1215/00127094-2018-0019