The voter model with a slow membrane

TL;DR

This paper studies a voter model on an infinite lattice with a slow membrane, analyzing its hydrodynamic limits and fluctuations, revealing how boundary conditions depend on the membrane's slowing parameter.

Contribution

It introduces a modified voter model with a slow membrane and characterizes its hydrodynamic behavior and fluctuation limits based on the membrane's parameters.

Findings

Hydrodynamic limit is a heat equation with boundary conditions depending on the slowing parameter.

Fluctuations are described by a generalized Ornstein-Uhlenbeck process with boundary conditions.

The model bridges microscopic dynamics and macroscopic boundary behaviors.

Abstract

We introduce the voter model on the infinite lattice with a slow membrane and investigate its hydrodynamic behavior and nonequilibrium fluctuations. 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 $\alpha N^{-\beta}$. Above, $\alpha>0,\,\beta \geq 0$ 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 $\beta$. For the nonequilibrium fluctuations, the limit is described by generalized Ornstein-Uhlenbeck process with certain boundary condition corresponding to the hydrodynamic equation.