Large-scale universal behaviour of Random Interfaces and Stochastic Operators

随机增长现象自然出现在各种物理和生物背景下，如燃烧前沿或细菌菌落的生长，薄膜上的晶体生长，湍流液晶等，即使所有这些现象可能会出现在微观尺度上非常不同，它们往往具有相同的大尺度行为，因此被认为属于同一个普适性类。这特别意味着，对描述这些大尺度行为的过程进行深入分析，必然会对同类中各种极其复杂的现实世界系统给出非常准确的定量和定性预测。在过去的40年里，数学界和物理界共同努力，确定了被广泛认为是唯一两个普遍过程，即Kardar-Parisi-Zhang和Edrwards-Wilkinson不动点，并研究了它们的普适性类。在最近的工作中，我建立了第三个，新的普遍性类的存在，完全错过了研究人员，并严格构建了其核心的普遍过程，布朗城堡。这个新课程的引入开启了许多新的刺激途径和一系列令人兴奋的问题，这些问题是本提案旨在调查和回答的。该研究计划的第二个支柱侧重于二维随机表面，从物理角度来看，这是特别相关的，因为它们对应于三维空间中二维表面的生长。尽管二维增长现象很重要，但它们是迄今为止最具挑战性和最不为人所知的。人们对它们的普适大尺度性质知之甚少，对涨落的更难的探索也几乎没有人探索过。本提案的目标是开发强有力的工具，以严格解决这些问题，从而为系统研究这些系统及其特点奠定基础。本研究计划的最后一个主题涉及安德森哈密顿量，也称为随机薛定谔算子。在这样的运营商的兴趣是出于它的分支连接到各种不同的数学和物理领域都从理论和应用的角度来看。事实上，安德森哈密顿量的谱性质与（随机）薛定谔方程的解理论或抛物安德森模型的性质、随机介质中的随机运动或随机环境中的分支过程有关。由于对安德森哈密顿量的普适性和著名的安德森局部化现象的研究，它引起了众多研究者的关注。该提案将通过用新技术补充现有文献，实现新的突破，并解决该领域长期存在的问题。

Stochastic growth phenomena naturally emerge in a variety of physical and biological contexts, such as growth of combustion fronts or bacterial colonies, crystal growth on thin films, turbulent liquid crystals, etc. Even though all these phenomena might appear very diverse at a microscopic scale, they often have the same large-scale behaviour and are therefore said to belong to the same Universality Class. This in particular means that an in-depth analysis of those processes describing these large-scale behaviours is bound to give very accurate quantitative and qualitative predictions about the wide variety of extremely complicated real-world systems in the same class. Over the last 40 years, the Mathematics and Physics communities in a joint effort determined what were widely believed to be the only two universal processes presumed to capture the large-scale behaviour of random interfaces in one spatial-dimension, namely the Kardar-Parisi-Zhang and Edrwards-Wilkinson Fixed Points, and studied their Universality Classes. In a recent work, I established the existence of a third, new universality class, entirely missed by researchers, and rigorously constructed the universal process at its core, the Brownian Castle. The introduction of this novel class opens a number of new stimulating pathways and a host of exciting questions that this proposal aims at investigating and answering. The second pillar of this research programme focuses on two-dimensional random surfaces, which are particularly relevant from a physical viewpoint as they correspond to the growth of two-dimensional surfaces in a three-dimensional space. Despite their importance, two-dimensional growth phenomena are by far the most challenging and the least understood. Very little is known concerning their universal large-scale properties and the even harder quest for fluctuations has barely been explored. The present proposal's goal is to develop powerful and robust tools to rigorously address these questions and consequently lay the foundations for a systematic study of these systems and their features. The last theme of this research plan concerns the Anderson Hamiltonian, also known as random Schrödinger operator. The interest in such an operator is motivated by its ramified connections to a variety of different areas in Mathematics and Physics both from a theoretical and a more applied perspective. Indeed, the spectral properties of the Anderson Hamiltonian are related to the solution theory of (random) Schrödinger's equations or properties of the parabolic Anderson model, random motion in random media or branching processes in random environment. The Anderson Hamiltonian has attracted the attention of a wide number of researchers, driven by the ambition of fully understanding its universal features and the celebrated phenomenon Anderson localisation. This proposal will establish new breakthroughs and tackle long-standing conjectures in the field by complementing the existing literature with novel techniques.