Large-scale universal behaviour of Random Interfaces and Stochastic Operators

随机生长现象自然出现在各种物理和生物环境中，如燃烧锋或细菌菌落的生长、薄膜上的晶体生长、湍流液晶等。尽管所有这些现象在微观尺度上可能看起来非常不同，但它们往往具有相同的大尺度行为，因此被称为属于同一普遍类。这尤其意味着，对描述这些大规模行为的那些过程的深入分析，必然会对同一类中各种极其复杂的现实世界系统做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仅有的两个普遍存在的过程，即Kardar-Parisi-Zhang和Edrwards-Wilkinson不动点，并研究了它们的普适性类别。在最近的一篇工作中，我证实了第三个新的普适性类的存在，而研究人员完全忽略了这一点，并严格构建了其核心--布朗城堡--的普适性过程。这一新颖课程的引入开启了一系列新的刺激途径和一系列令人兴奋的问题，本提案旨在调查和回答这些问题。这一研究方案的第二个支柱侧重于二维随机表面，从物理角度来看，这一点特别相关，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，但到目前为止，它们是最具挑战性的，也是最不被理解的。人们对它们的普遍大规模性质知之甚少，而对波动更难的探索也几乎没有人探索过。本提案的目标是开发强大而有力的工具，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。这项研究计划的最后一个主题涉及安德森哈密顿，也被称为随机薛定谔算子。人们对这种运算符的兴趣是因为它与数学和物理学中不同领域的分支联系，从理论和更实用的角度来看都是如此。实际上，Anderson哈密顿量的谱性质与(随机)薛定谔方程的解理论或抛物型Anderson模型、随机介质中的随机运动或随机环境中的分支过程的性质有关。安德森哈密尔顿理论吸引了大量研究人员的注意，其动力来自于充分理解其普遍特征和著名的安德森本土化现象的野心。这一提议将通过用新的技术补充现有文献，建立新的突破，并解决该领域长期存在的猜测。

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.