Turing Instability in a Model with Two Interacting Ising Lines: 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 equations obtained in Capanna and Soprano (Markov Proc Relat Fields 23(3):401-420, 2017), we find conditions under which Turing instability occurs around the zero 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 time defined as the one at which the process starts to be of finite order, we prove that the +/- 1-Fourier modes are uniformly away from zero.