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Large-scale universal behaviour of Random Interfaces and Stochastic Operators

随机接口和随机算子的大规模通用行为

基本信息

批准号：
MR/W008246/1
负责人：
Giuseppe Cannizzaro
金额：
$ 90.67万
依托单位：
University of Warwick
依托单位国家：
英国
项目类别：
Fellowship
财政年份：
2022
资助国家：
英国
起止时间：
2022 至无数据
项目状态：
未结题

来源：
https://gtr.ukri.org/projects?ref=MR%2FW008246%2F1
关键词：
Large scale universal behaviour Random

项目摘要

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.

随机生长现象自然出现在各种物理和生物环境中，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。尽管所有这些现象在微观尺度上可能表现得非常不同，但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世性类。这尤其意味着，对描述这些大规模行为的过程进行深入分析，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的过程，即卡尔达-帕里西-张和爱德华-威尔金森不动点，并研究了它们的普遍性类。在最近的一篇文章中，我确立了第三个新的普世性类别的存在，这一类别完全被研究人员所忽视，并在其核心——布朗城堡——严格地构建了普世性过程。这门新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。本研究计划的第二个支柱集中在二维随机表面，这是特别相关的，从物理角度来看，因为它们对应于二维表面在三维空间的增长。尽管它们很重要，但二维生长现象是迄今为止最具挑战性和最不为人所知的。关于它们普遍的大尺度性质，我们所知甚少，而对波动的更难的探索几乎没有被探索过。本提案的目标是开发强大而稳健的工具来严格解决这些问题，从而为系统地研究这些系统及其特征奠定基础。本研究计划的最后一个主题涉及安德森哈密顿算子，也称为随机Schrödinger算子。从理论和应用的角度来看，这种算子与数学和物理中各种不同领域的分支联系激发了人们对这种算子的兴趣。事实上，安德森哈密顿量的谱性质与（随机）Schrödinger方程的解理论或抛物型安德森模型的性质、随机介质中的随机运动或随机环境中的分支过程有关。安德森-汉密尔顿函数吸引了众多研究者的注意，他们想要充分理解它的普遍特征和著名的安德森局域化现象。本提案将建立新的突破，并通过补充现有文献和新技术来解决该领域长期存在的猜想。