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

Guillaume Barraquand, Alexei Borodin, Ivan Corwin, Michael Wheeler

Published 2017-04-14Version 1

We study the asymmetric simple exclusion process (ASEP) on the positive integers with open boundary condition. We show that if the right jump rate is $1$, the left jump rate is $t\in (0,1)$, and there is a reservoir of particles at the origin that injects particles at rate $1/2$ and ejects particles at rate $t/2$, then when starting devoid of particles, the number of particles in the system at time $\tau$ fluctuates according to the Tracy-Widom GOE distribution on the $\tau^{1/3}$ scale. We also study the convergence of the height function in the weak asymmetry scaling, where the height profile is expected to converge to the KPZ equation with Neumann type boundary condition. Our main tool is a new class of probability measures on Young diagrams that we call half-space Macdonald processes. Although these measures do not define Pfaffian point processes in general, we show that the expectation of certain multiplicative functionals of the process can be expressed as Fredholm Pfaffians. In the case where Macdonald functions degenerate to Hall-Littlewood functions, the measure is related to an inhomogeneous stochastic six-vertex model in a half-quadrant. The latter model rescales to half-line ASEP, which yields exact formulas for the distribution of the height function at the origin.