Evolution of the Interfaces in a Two Dimensional Potts Model

We investigate the evolution of the random interfaces in a two dimensional Potts model at zero temperature under Glauber dynamics for some particular initial conditions. We prove that under space-time diffusive scaling the shape of the interfaces converges in probability to the solution of a non-linear parabolic equation. This Law of Large Numbers is obtained from the Hydrodynamic limit of a coupling between an exclusion process and an inhomogeneous one dimensional zero range process with asymmetry at the origin. 1. The Potts model We consider a 3-state Potts model at zero temperature under Glauber dynamics described by a spin system on Ωsp = {−1, 0, 1}Z 2 , whose generator acting on cylinder functions is given by (Lspf)(σ) = 1 2 ∑