Abstract
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.
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Acknowledgements
Zhao thanks the financial support from the Fundamental Research Funds for the Central Universities in China and from the ANR grant MICMOV (ANR-19-CE40-0012) of the French National Research Agency (ANR). Xue thanks the financial support from the National Natural Science Foundation of China with grant number 12371142 and the Fundamental Research Funds for the Central Universities with grant number 2022JBMC039.
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Zhao proved the hydrodynamic limit theorem. Xue proved the fluctuation theorem.
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Zhao, L., Xue, X. The Voter Model with a Slow Membrane. J Theor Probab (2024). https://doi.org/10.1007/s10959-024-01321-9
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DOI: https://doi.org/10.1007/s10959-024-01321-9