  The one-dimensional totally asymmetric simple exclusion process (TASEP), a
Markov process describing classical hard-core particles hopping in the same
direction, is considered on a periodic lattice of $L$ sites. The relaxation to
the non-equilibrium steady state, which occurs on the time scale $t\sim
L^{3/2}$ for large $L$, is studied for the half-filled system with $N=L/2$
particles. Using large $L$ asymptotics of Bethe ansatz formulas for the
eigenstates, exact expressions depending explicitly on the rescaled time
$t/L^{3/2}$ are obtained for the average and two-point function of the local
density, and for the current fluctuations for simple (stationary, flat and
step) initial conditions, relating previous results for the infinite system to
stationary large deviations. The final formulas have a nice interpretation in
terms of a functional integral with the action of a scalar field in a linear
potential.
