2016/10/31 by Orr, Daniel, Petrov, Leonid
#Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Quantum Algebra (math.QA)
paper · doi:10.48550/arxiv.1610.10080
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 \mathfrakh in the higher spin six vertex model first computed in arXiv:1601.05770 using Bethe ansatz. This result in particular implies that for the vertex model with the step Bernoulli boundary condition, the value of \mathfrakh 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 \mathfrakh(N+1,T) and xN+N in distribution. Combined with the identification of averages of observables between the stochastic higher spin six vertex model and Schur measures (which are t=q Macdonald measures) obtained recently in arXiv:1608.01553, this produces GUE Tracy--Widom asymptotics for a discrete time q-TASEP with the step initial configuration and special jump parameters.