Multi-point distribution of discrete time periodic TASEP

Abstract

We study the one-dimensional discrete time totally asymmetric simple exclusion process with parallel update rules on a spatially periodic domain. A multi-point space-time joint distribution formula is obtained for general initial conditions. The formula involves contour integrals of Fredholm determinants with kernels acting on certain discrete spaces. The large-time, large-period limit is obtained under the relaxation time scale t = O(L 3/2 ) when the height fluctuations are critically affected by the finite geometry for a class of initial conditions satisfying certain technical assumptions. The assumptions are verified for step and flat initial conditions. As a corollary we obtain the multi-point distribution for discrete time TASEP on the whole integer lattice Z by taking the period L large enough so that the finite-time distribution is not affected by the boundary. The large time limit for multi-time distribution for discrete time TASEP on Z is also obtained for step initial condition.