  Using theories of phase ordering kinetics and of renormalization group, we
derive analytically the relaxation times of the long wave-length fluctuations
of a phase-separated domain boundary in the vicinity of (and below) the
critical temperature, in the planar Ising universality class. For a conserved
order parameter, the relaxation time grows like $\Lambda^3$ at wave-length
$\Lambda$ and can be expressed in terms of parameters relevant at the
microscopic scale: lattice spacing, bulk diffusion coefficient of the minority
phase, and temperature. These results are supported by numerical simulations of
2D Ising models, enabling in addition to calculate the non-universal numerical
prefactor. We discuss the applications of these findings to the determination
of the real time-scale associated with elementary Monte Carlo moves from the
measurement of long wave-length relaxation times on experimental systems or
Molecular Dynamics simulations.
