Curve Shortening Flow for Spatial Random Permutations

Motivated by mean-curvature type droplet shrinking in the hydrodynamic scaling limit for the Ising model on a 2-dimensional lattice at zero temperature, we analyse a similar microscopic model based on spatial random permutations. This naturally leads to interacting particle systems and the derivation of their hydrodynamic equations. In the progress, we deal with non-gradient systems and non-product stationary measures.