The voter model with a slow membrane

Abstract

We introduce the voter model on the infinite lattice with a slow membrane and investigate its hydrodynamic behavior. The model is defined as follows: a voter adopts one of its neighbors' opinion at rate one except for neighbors crossing the hyperplane {x : x 1 = 1/2}, where the rate is αN −β . Above, α > 0, β ≥ 0 are two parameters and N is the scaling parameter. The hydrodynamic equation turns out to be heat equation with various boundary conditions depending on the value of β. The proof is based on duality method.