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 \(\{0,1\}^{\mathbb Z^d}\) ${\{0,1\}}^{{\mathbb{Z}}^d}$ . In our model, a voter adopts one of its neighbors’ opinion at rate one except for neighbors crossing the hyperplane \(\{x:x_1 = 1/2\}\) $\{x:x_1=1/2\}$ , where the rate is \(\alpha N^{-\beta }\) $\alpha N^{-\beta }$ and thus is called a slow membrane. Above, \(\alpha >0 \ \textrm{and} \ \beta \ge 0\) $\alpha >0\text{and}\beta \ge 0$ are given parameters and the positive integer N is a scaling parameter. We consider the limit \(N \rightarrow \infty \) $N\to \infty$ and prove that the hydrodynamic limits are given by the heat equation without or with Robin/Neumann conditions depending on the values of \(\beta \) $\beta$ . 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.