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 } 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 $$\alpha N^{-\beta }$$ α N - β and thus is called a slow membrane. Above, $$\alpha >0 \ \textrm{and} \ \beta \ge 0$$ α > 0 and β ≥ 0 are given parameters and the positive integer N is a scaling parameter. We consider the limit $$N \rightarrow \infty $$ N → ∞ and prove that the hydrodynamic limits are given by the heat equation without or with Robin/Neumann conditions depending on the values of $$\beta $$ β . 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.