The emergence of universal behaviour for growth models, stochastic PDEs and random operators.

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In 1986, three physicists, Kardar, Parisi and Zhang, conjectured that all randomly evolving surfaces possessing three features, a smoothing mechanism, an underlying locally uncorrelated noise and a growth mechanism depending on the size of the slope, should have the same large-scale fluctuations, irrespective of their microscopic details. In other words, they predicted the existence of a Universality Class, that since then bares their name, and of a universal stochastic process, able to capture the behaviour of a wide class of models, such as turbulent liquid crystals, crystal growth on thin films, bacteria colony growth, etc. Over the last thirty years, their work stimulated the interest of a wide number of researchers, driven by the ambition to fully understand the nature of the KPZ Universality Class and to characterise this universal object. On the other hand, the Physics literature also predicts that, when a physical system possesses the same features apart from the slope dependence, then it belongs to a different Universality Class, the so-called Edwards-Wilkinson (EW) Universality Class, named after the two physicists that introduced it, and the universal process describing their behaviour is Gaussian and can be easily explicitly characterised. The first objective of this research proposal is to show that in the context of (1+1)-dimensional (one for time and one for space) randomly evolving interfaces, the classification given above is not exhaustive and another Universality Class needs to be considered. Our goal is to rigorously construct the universal object at its core, a stochastic process called Growing Brownian Castle, determine its characterising properties, give the first instances of its universality and analyse its relation with KPZ. In the context of the KPZ Universality Class, there is a model that plays a distinguished role and it is presumed to be universal itself. This model is a Stochastic Partial Differential Equation (SPDE), the KPZ Equation. Despite its importance, a satisfactory solution theory for this equation in one spatial dimension was established only recently thanks to the theory of Regularity Structures, by M. Hairer. The techniques that are now available allow for a systematic study of its universality and this research program intends to establish it for a family of models driven by conservative dynamic, which has never been considered so far.For evolving surfaces in (1+2)-dimensions, the Universality Classes picture is subtler because the slope can evolve in different directions that could compete with each other. This proposal focuses on the case in which the contribution of the slope sizes in the different directions averages out. This class of models is called Anisotropic KPZ Universality Class and the long-standing conjecture, coming from the Physics literature, is that this class is nothing but EW in dimension 2. In other words it is expected that the slope does not play any role at all. The project aims at showing such a result for the Anisotropic KPZ Equation, a singular SPDE that cannot be treated by the theory of Regularity Structures mentioned above and for which radically new ideas are needed. At last, the random operator we will focus on is the Anderson-Hamiltonian. Its importance lies on the fact that it is connected with the parabolic Anderson model, the scaling limit of random motion in random potential or branching processes in random media, and many others. We will determine some of its properties that will shed some light on its universal nature.