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

1986年，Kardar、Parisi和Zhang三位物理学家推测，所有随机演化的表面都具有三个特征：平滑机制、潜在的局部不相关噪声和依赖于斜率大小的生长机制，无论其微观细节如何，它们都应该具有相同的大规模波动。换句话说，他们预测了一个普适性类的存在，从那时起他们的名字就出现了，还有一个普适性随机过程，能够捕捉到各种模型的行为，比如湍流液晶、薄膜上的晶体生长、细菌菌落生长等。在过去的三十年里，他们的工作激发了大量研究人员的兴趣，他们的目标是充分了解KPZ普遍性类的性质，并描述这个普遍对象的特征。另一方面，物理学文献也预测，当一个物理系统除了斜率依赖之外具有相同的特征时，那么它属于不同的普惠类，即所谓的爱德华兹-威尔金森（EW）普惠类，以引入它的两位物理学家命名，描述其行为的普惠过程是高斯的，可以很容易地明确表征。本研究方案的第一个目标是表明在（1+1）维（一个为时间，一个为空间）随机演化接口的情况下，上述分类并不详尽，需要考虑另一个普适性类。我们的目标是严格构建其核心的普遍对象，一个称为生长布朗城堡的随机过程，确定其特征属性，给出其普遍性的第一个实例，并分析其与KPZ的关系。在KPZ普适性类的背景下，有一个模型起着显著的作用，它本身被认为是普适性的。这个模型是一个随机偏微分方程（SPDE），即KPZ方程。尽管它很重要，一个令人满意的一维空间方程的解理论直到最近才建立起来，这要归功于M. Hairer的规则结构理论。目前可用的技术允许对其普遍性进行系统的研究，本研究计划打算为保守动态驱动的一系列模型建立它，这是迄今为止从未考虑过的。对于（1+2）维的进化曲面，普适类图更加微妙，因为斜率可以向不同的方向进化，这些方向可能相互竞争。这一建议的重点是在不同方向的坡度大小的贡献平均的情况下。这类模型被称为各向异性KPZ通用性类，来自物理文献的长期猜想是，这类模型只不过是2维的EW。换句话说，我们期望斜率不起任何作用。该项目旨在为各向异性KPZ方程展示这样的结果，这是一个奇异的SPDE，不能用上面提到的规则结构理论来处理，需要全新的思想。最后，我们将重点讨论的随机算子是安德森-哈密顿算子。它的重要性在于它与抛物型安德森模型、随机势或随机介质中分支过程中随机运动的标度极限以及其他许多问题相联系。我们将确定它的一些特性，这些特性将揭示它的普遍性质。

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.