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

1986年，三位物理学家Kardar、Parisi和Zhang推测，所有随机演化的表面都应该具有三个特征，即平滑机制、潜在的局部不相关噪声和取决于坡度大小的增长机制，无论它们的微观细节如何。换句话说，他们预测了一个普适类的存在，从那时起他们就暴露了自己的名字，以及一个普适随机过程的存在，能够捕捉到一大类模型的行为，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，驱使他们充分了解KPZ普适类的性质并描述这个普适对象。另一方面，物理学文献还预测，当一个物理系统除了斜率依赖性之外具有相同的特征时，它属于一个不同的普适性类，即所谓的爱德华兹-威尔金森(EW)普适性类，以两位提出它的物理学家的名字命名，描述它们行为的普适过程是高斯的，可以很容易地明确地刻画出来。本研究的第一个目标是证明在(1+1)维(一个用于时间，一个用于空间)随机演化界面的背景下，上面给出的分类不是详尽的，需要考虑另一个普适性类。我们的目标是严格地构造其核心的普适对象，一个称为成长布朗城堡的随机过程，确定它的特征性质，给出它的普适性的第一个实例，并分析它与KPZ的关系。在KPZ普遍性类别的背景下，有一种模式发挥着独特的作用，并被推定为普遍模式。该模型是一个随机偏微分方程(SPDE)，即KPZ方程。尽管它很重要，但直到最近，由于M.Hairer的正则性结构理论，才建立了一维空间中该方程的满意解理论。目前可用的技术允许对其普适性进行系统的研究，该研究计划旨在为保守动力学驱动的一族模型建立普适性，这是迄今为止从未考虑过的。对于(1+2)维的演化曲面，普适性类别图更为微妙，因为坡度可以向不同的方向演化，从而相互竞争。这一建议侧重于不同方向上的坡度大小的贡献平均的情况。这类模型被称为各向异性KPZ普适类，来自物理学文献的长期猜想是，这类模型只不过是2维的EW。换句话说，人们认为斜率根本不起任何作用。该项目旨在展示各向异性KPZ方程的这种结果，这是一种奇异的SPDE，不能用上面提到的正则性结构理论来处理，需要根本上的新想法。最后，我们将关注的随机算子是安德森-哈密顿算符。它的重要性在于它与抛物型Anderson模型、随机势中随机运动的标度极限或随机介质中的分支过程等许多事实有关。我们将确定它的一些性质，这些性质将有助于揭示它的普遍性质。

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.