  We study the long time asymptotics of the relaxation dynamics of the totally
asymmetric simple exclusion process on a ring. Evaluating the asymptotic
amplitudes of the local currents by the algebraic Bethe ansatz method, we find
the relaxation times starting from the step and alternating initial conditions
are governed by different eigenvalues of the Markov matrix. In both cases, the
scaling exponents of the leading asymptotic amplitudes with respect to the
total number of sites are found to be -1. We also study the asymptotics of
correlation functions such as the emptiness formation probability.
