Article ID Journal Published Year Pages File Type
5778402 Advances in Mathematics 2017 53 Pages PDF
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].
Related Topics
Physical Sciences and Engineering Mathematics Mathematics (General)
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