  The dynamic behavior of the interfaces in the standard and random driven
lattice gas models (DLG and RDLG respectively) is investigated via numerical
Monte Carlo simulations in two dimensions. For $T\lesssim T_c$, the average
interface width of the strips ($W$) was measured as a function of the lattice
size and the anisotropic shape factor. It was found that the saturation value
$W^{2}_{sat}$ only depends on the lattice size parallel to the external field
axis $L_y$ and exhibits two distinct regimes: $W^{2}_{sat}\propto \ln$ $L_y$
for low temperatures, that crosses over to $W^{2}_{sat}\propto L_y^{2\alpha_I}$
near the critical zone, $\alpha_I=1/2$ being the roughness exponent of the
interface. By using the relationship $\alpha_I=1/(1+\Delta_I)$, the anisotropic
exponent for the interface of the DLG model was estimated, giving
$\Delta_I\simeq 1$, in agreement with the computed value for anisotropic bulk
exponent $\Delta_B$ with a recently proposed theoretical approach. At the
crossover region between both regimes, we observed indications of bulk
criticality. The time evolution of $W$ at $T_c$ was also monitored and shows
two growing stages: first one observes that $W \propto \ln$ $t$ for several
decades, and in the following times one has $W\propto t^{\beta_I}$, where
$\beta_{I}$ is the dynamic exponent of the interface width. By using this value
we estimated the dynamic critical exponent of the correlation length in the
perpendicular direction to the external field, giving $z_{\perp}^I\approx 4$,
which is consistent with the dynamic exponent of the bulk critical transition
$z_{\perp}^B$ in both theoretical approaches developed for the standard model.
A similar scenario was also observed in the RDLG model, suggesting that both
models may belong to the same universality class.
