  We investigate numerically the relaxation dynamics of an elastic string in
two-dimensional random media by thermal fluctuations starting from a flat
configuration. Measuring spatial fluctuations of its mean position, we find
that the correlation length grows in time asymptotically as $\xi \sim (\ln
t)^{1/\tilde\chi}$. This implies that the relaxation dynamics is driven by
thermal activations over random energy barriers which scale as $E_B(\ell) \sim
\ell^{\tilde\chi}$ with a length scale $\ell$. Numerical data strongly suggest
that the energy barrier exponent $\tilde{\chi}$ is identical to the energy
fluctuation exponent $\chi=1/3$. We also find that there exists a long
transient regime, where the correlation length follows a power-law dynamics as
$\xi \sim t^{1/z}$ with a nonuniversal dynamic exponent $z$. The origin of the
transient scaling behavior is discussed in the context of the relaxation
dynamics on finite ramified clusters of disorder.
