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We study the periodic $q$-Whittaker and Hall-Littlewood processes, twoprobability measures on sequences of partitions. We prove that a certainobservable of the periodic $q$-Whittaker process exhibits a $(q,u)$ symmetryafter a random shift, generalizing a previous result of Imamura, Mucciconi, andSasamoto who showed a matching between the periodic Schur and $q$-Whittakermeasures, and also give a vertex model formulation of their result. As part ofour proof of the $(q,u)$ symmetry, we obtain contour integral formulas for boththe periodic $q$-Whittaker and Hall-Littlewood processes. We also show amatching between certain observables in the periodic Hall-Littlewood processand in a quasi-periodic stochastic six vertex model after a suitable randomshift, and discuss a limit to the stationary periodic stochastic six vertexmodel.

Periodic $q$-Whittaker and Hall-Littlewood processes