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

增长模型、随机偏微分方程和随机算子的通用行为的出现。

基本信息

批准号：
EP/S012524/1
负责人：
Giuseppe Cannizzaro
金额：
$ 45万
依托单位：
University of Warwick
依托单位国家：
英国
项目类别：
Fellowship
财政年份：
2018
资助国家：
英国
起止时间：
2018 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FS012524%2F1
关键词：
emergence universal behaviour growth models

项目摘要

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.

在1986年，三位物理学家Kardar、Parisi和Zhang指出，所有随机演化的表面都具有三个特征，即平滑机制、潜在的局部不相关噪声和依赖于斜率大小的增长机制，无论它们的微观细节如何，都应该具有相同的大尺度波动。换句话说，他们预测了一个普遍性类的存在，从那时起，他们的名字就暴露了，还有一个普遍的随机过程，能够捕捉到广泛的一类模型的行为，例如湍流液晶，薄膜上的晶体生长，细菌菌落生长等。我们的目标是充分理解KPZ普遍性课程的本质，并实现这一普遍目标。另一方面，物理学文献也预言，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-威尔金森（EW）普适性类，以两位物理学家的名字命名，描述它们行为的普适过程是高斯的，可以很容易地明确表征。本研究建议的第一个目标是表明，在（1+1）维（一个时间和空间）随机演化界面的背景下，上述分类是不详尽的，需要考虑另一个普适类。我们的目标是严格地构建其核心的普遍对象，一个随机过程称为增长布朗城堡，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普遍性类的上下文中，有一个模型起着杰出的作用，它被认为是普遍的。该模型是一个随机偏微分方程（SPDE），即KPZ方程。尽管它的重要性，一个令人满意的解决方案理论，这个方程在一个空间维是最近才建立的正则结构理论，由M。海尔现有的技术允许系统地研究它的普适性，本研究计划旨在建立一个由保守动力学驱动的模型族，这是迄今为止从未考虑过的。对于（1+2）维的演化表面，普适性类图像是微妙的，因为斜率可以在不同的方向上演化，可以相互竞争。该建议侧重于不同方向上的坡度大小的贡献达到平均值的情况。这类模型被称为各向异性KPZ普遍性类，来自物理文献的长期猜测是，这类模型只不过是2维的EW。换句话说，预计斜率根本不起任何作用。该项目旨在展示各向异性KPZ方程的结果，这是一个奇异的SPDE，不能用上面提到的正则结构理论来处理，需要全新的想法。最后，我们将重点讨论的随机算子是Anderson-Hamilton算子。它的重要性在于它与抛物线安德森模型、随机势中随机运动的标度极限或随机介质中的分支过程以及许多其他模型有关。我们将确定它的一些性质，这些性质将揭示它的普遍性质。