The Voter Model with a Slow Membrane

We introduce the voter model on the infinite integer lattice with a slow membrane and investigate its hydrodynamic behavior and nonequilibrium fluctuations. The voter model is one of the classical interacting particle systems with state space \(\{0,1\}^{\mathbb Z^d}\). In our model, 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 \(\alpha N^{-\beta }\) and thus is called a slow membrane. Above, \(\alpha >0 \ \textrm{and} \ \beta \ge 0\) are given parameters and the positive integer N is a scaling parameter. We consider the limit \(N \rightarrow \infty \) and prove that the hydrodynamic limits are given by the heat equation without or with Robin/Neumann conditions depending on the values of \(\beta \). We also consider the nonequilibrium fluctuations, where the limit is described by generalized Ornstein–Uhlenbeck processes with certain boundary conditions corresponding to the hydrodynamic equation.