Final Report Summary - CRITICAL (Behaviour near criticality) One of the guiding principles of modern theoretical physics is that of "universality": the large-scale behaviour of random systems often depends only on very few features of the complex microscopic components which constitute it. A natural class of such universal random objects are "renormalisation fixed points" which are invariant under the operation of "zooming to larger scales". Unfortunately, only relatively few of these objects can be characterised and these have usually some simplifying features: Gaussianity, one-dimensionality, conformal invariance, integrability. A somewhat simpler class of universal objects consists of "crossover regimes". In these regimes, one considers objects that themselves are not scale invariant, but that converge to two different renormalisation fixed points at small and large scales. In situations where the small-scale behaviour is Gaussian, these crossover regimes are often described by a singular stochastic partial differential equation. These equations are themselves interesting objects of study due to their ill-posed character: while it is often relatively straightforward to derive expressions for them, these are typically non-sensical when taken at face-value due to the appearance of nonlinearities involving very singular terms. One outcome of this project was to provide a framework for a systematic description of these objects as well as the study of their universal properties. We made important progress in this direction by building a general theory which provides a canonical notion of solution for a large class of ill-posed equations and shows that this notion is surprisingly stable under a very large class of perturbations. We also started building a framework allowing to study the convergence of discrete systems to these solutions which opens the door to linking the theory back to classical models of statistical mechanics. We also obtained a number of results providing a toolbox to further analyse the fine properties of these solutions.
