Publication date:
7 September 2017
Source:Advances in Mathematics, Volume 317
Author(s): Daniel Orr, Leonid Petrov
We present two new connections between the inhomogeneous stochastic higher spin six vertex model in a quadrant and integrable stochastic systems from the Macdonald processes hierarchy. First, we show how Macdonald q-difference operators with $t=0$ (an algebraic tool crucial for studying the corresponding Macdonald processes) can be utilized to get q-moments of the height function $\mathfrak{h}$ in the higher spin six vertex model first computed in [21] using Bethe ansatz. This result in particular implies that for the vertex model with the step Bernoulli boundary condition, the value of $\mathfrak{h}$ at an arbitrary point $(N+1,T)\in {\mathbb{Z}}_{\ge 2}\times {\mathbb{Z}}_{\ge 1}$ has the same distribution as the last component ${\lambda }_N$ of a random partition under a specific $t=0$ Macdonald measure. On the other hand, it is known that ${\mathbf{x}}_N:={\lambda }_N-N$ can be identified with the location of the Nth particle in a certain discrete time q-TASEP started from the step initial configuration. The second construction we present is a coupling of this q-TASEP and the higher spin six vertex model (with the step Bernoulli boundary condition) along time-like paths providing an independent probabilistic explanation of the equality of $\mathfrak{h}(N+1,T)$ and ${\mathbf{x}}_N+N$ in distribution. As an illustration of our main results we obtain GUE Tracy–Widom asymptotics of a certain discrete time q-TASEP (with the step initial configuration and special jump parameters) by means of Schur measures (which are $t=q$ Macdonald measures). This analysis combines our results with the identification of averages of observables between the stochastic higher spin six vertex model and Schur measures obtained recently in [8].