Free field theory and observables of periodic Macdonald processes

Abstract: We propose periodic Macdonald processes as a $(q,t)$-deformation of periodic Schur processes and a periodic analogue of Macdonald processes. It is known that, in the theory of stochastic processes related to a family of symmetric functions, the Cauchy-like identity gives an explicit expression of a partition function. We compute the partition functions of periodic Macdonald processes relying on the free field realization of the Macdonald theory. We also study several families of observables for periodic Macdonald processes and give formulas of their moments. The technical tool is the free field realization of operators that are diagonalized by the Macdonald symmetric functions, where the operators admit expressions by means of vertex operators. We show that, when we adopt Plancherel specializations, the corresponding periodic Macdonald process is related to a Young diagram-valued periodic continuous process. It is known that, for periodic Schur processes, determinantal formulas are only available when we extend them to take into account the charge. We also consider this kind of shift-mixed periodic Macdonald processes and discuss their Schur-limit.