  The relaxation time limit of the one-point distribution of the spatially
periodic totally asymmetric simple exclusion process is expected to be the
universal one point distribution for the models in the KPZ universality class
in a periodic domain. Unlike the infinite line case, the limiting one point
distribution depends non-trivially on the scaled time parameter. We study
several properties of this distribution for the case of the periodic step and
flat initial conditions. We show that the distribution changes from a
Tracy-Widom distribution in the small time limit to the Gaussian distribution
in the large time limit, and also obtain right tail estimate for all time.
Furthermore, we establish a connection to integrable differential equations
such as the KP equation, coupled systems of mKdV and nonlinear heat equations,
and the KdV equation.
