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We present a general method for constructing integrable stochastic processes,with two-step discrete time Floquet dynamics, from the transfer matrixformalism. The models can be interpreted as a discrete time parallel update.The method can be applied for both periodic and open boundary conditions. Wealso show how the stationary distribution can be built as a matrix productstate. As an illustration we construct a parallel discrete time dynamicsassociated with the R-matrix of the SSEP and of the ASEP, and provide theassociated stationary distributions in a matrix product form. We use thisgeneral framework to introduce new integrable generalized exclusion processes,where a fixed number of particles is allowed on each lattice site in oppositionto the (single particle) exclusion process models. They are constructed usingthe fusion procedure of R-matrices (and K-matrices for open boundaryconditions) for the SSEP and ASEP. We develop a new method, that we named"fused" matrix ansatz, to build explicitly the stationary distribution in amatrix product form. We use this algebraic structure to compute physicalobservables such as the correlation functions and the mean particle current.

Integrable Floquet dynamics, generalized exclusion processes and "fused" matrix ansatz