TASEP on a ring in sub-relaxation time scale

Interacting particle systems in the KPZ universality class on a ring of size $L$ with $O(L)$ number of particles are expected to change from KPZ dynamics to equilibrium dynamics at the so-called relaxation time scale $t=O(L^{3/2})$. In particular the system size is expected to have little effect to the particle fluctuations in the sub-relaxation time scale $t\ll L^{3/2}$. We prove that this is indeed the case for the totally asymmetric simple exclusion process (TASEP) with two types of initial conditions. For flat initial condition, we show that the particle fluctuations are given by the Airy$_1$ process as in the infinite TASEP with flat initial condition. On the other hand, the TASEP on a ring with step initial condition is equivalent to the periodic TASEP with a certain shock initial condition. We compute the fluctuations explicitly both away from and near the shocks for the infinite TASEP with same initial condition, and then show that the periodic TASEP has same fluctuations in the sub-relaxation time scale.