Hydrodynamical Limits and Geometric Measure Theory: Mean Curvature Limits from a Threshold Voter Model

We consider hydrodynamical limits for a simple threshold voter model for a microscopically evolving random interface. This model, which is a zero-temperature Ising model, was studied by Spohn in a 1+1 setting. The model leads to motion by a certain anisotropic mean curvature. Here we develop this model through some notions of geometric measure theory, dispensing with the 1+1 restriction.