Turing instability in a model with two interacting Ising lines: linear stability and non-equilibrium fluctuations

This is the second of two articles on the study of a particle system model that exhibits a Turing instability type effect. About the hydrodynamic quations obtained in \cite{CSL17a}, we find conditions under which Turing instability occurs around the null equilibrium solution. In this instability regime: for long times at which the process is of infinitesimal order, we prove that the non-equilibrium fluctuations around the hydrodynamic limit are Gaussian; for times converging to the critical one at which the process is of finite order, we prove that the $\pm 1$-Fourier modes are uniformly away from zero.