  We study large deviations for a Markov process on curves in $\mathbb{Z}^2$
mimicking the motion of an interface. Our dynamics can be tuned with a
parameter $\beta$, which plays the role of an inverse temperature, and
coincides at $\beta$ = $\infty$ with the zero-temperature Ising model with
Glauber dynamics, where curves correspond to the boundaries of droplets of one
phase immersed in a sea of the other one. We prove that contours typically
follow a motion by curvature with an influence of the parameter $\beta$, and
establish large deviations bounds at all large enough $\beta$ < $\infty$. The
diffusion coefficient and mobility of the model are identified and correspond
to those predicted in the literature.
