Stochastic higher spin six vertex model and q-TASEPs

Abstract

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 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 h at an arbitrary point (N+1,T)∈Z≥2×Z≥1 has the same distribution as the last component λN of a random partition under a specific t=0 Macdonald measure. On the other hand, it is known that xN:=λN−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 h(N+1,T) and xN+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].