Free field theory and observables of periodic Macdonald processes

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