Motion and pinning of discrete interfaces

We describe the motion of interfaces in a two-dimensional discrete environment by coupling the minimizing movements approach by Almgren, Taylor and Wang and a discrete-to-continuous analysis. We show that below a critical ratio of the time and space scalings we have no motion of interfaces (pinning), while above that ratio the discrete motion is approximately described by the crystalline motion by curvature on the continuum described by Almgren and Taylor. The critical regime is much richer, exhibiting a pinning threshold (small sets move, large sets are pinned), partial pinning (portions of interfaces may not move), pinning after an initial motion (possibly to a non-convex limit set), “quantization” of the interface velocity, and non-uniqueness effects.