Convergence of nonlocal threshold dynamics approximations to front propagation

In this note we prove that appropriately scaled threshold dynamics-type algorithms corresponding to the fractional Laplacian of order $\alpha \in (0,2)$ converge to moving fronts. When $\alpha \geqq 1$ the resulting interface moves by weighted mean curvature, while for $\alpha <1$ the normal velocity is nonlocal of ``fractional-type.'' The results easily extend to general nonlocal anisotropic threshold dynamics schemes.