When is spontaneous formation of spatial patterns robust?

From atomic spins in magnets to galaxies, and from embryonic development to patchy vegetation in arid environments, the physical world is filled with complex systems that spontaneously form spatial patterns. While simple mathematical models, such as reaction-diffusion systems, can explain the formation of such patterns, the complexity of the physical systems these models aim to describe necessitates the analysis of model robustness. Our work utilizes random matrix theory to provide arguably the first definition of robustness that is analytically tractable, providing an easy guide for identifying spatial interactions that robustly generate spatial patterns. We illustrate our theory on examples from mathematical biology, showing that diffusion alone cannot robustly generate spatial Turing patterns in large and unstructured systems, while advection, chemotaxis and non-local interactions can robustly promote pattern formation. Furthermore, we use our theory to prove that the spinodal decompositon of soft condensed matter is dynamically robust and predict the dynamics beyond regimes permitted by the standard Landau-Ginzburg theory. By classifying different spatial interactions based on the robustness of pattern formation, this work provides insights into which mechanisms are fundamental for pattern formation in large and unstructured physical systems.