Connectivity functions for random walk percolation models

从历史上看，研究流体（水、污染物等）在介质中的扩散现象具有极大的兴趣和实际意义，为此引入了许多数学模型。M. Sahimi（1994）将这类模型分为两类：扩散过程和渗透过程。在第一种假设中，随机性的来源在于流体，而在第二种假设中，流体的运动完全由介质的结构决定，而介质本身被认为是随机产生的。S. R. Broadbent和J. M. Hammersley（1957）最初对渗透过程所采取的观点进行了研究和背景化，并从此导致了对各种随机几何的进一步探索，如伯努利渗透模型或高斯自由场（GFF）偏移集。后者产生了新的和令人兴奋的普适性类，其特点是局部可观测值之间存在长程相关性，其随距离多项式衰减。所有这些模型都表现出所谓的相变，可能是二阶的，并以临界指数为特征，它发生在密度从仅由小岛屿组成的亚临界相转变为由无限簇组成的超临界相时。本研究项目的目标是进一步发展我们对远程相关渗流模型的数学理解。整数格上的一种这样的模型被称为随机交错，最早是由a - s引入的。Sznitman（2010）处理随机游走轨迹的概率覆盖和碎片化问题。例如，M.J. Brummelhuis和H.J. Hillhorst（1991）认为这些模型是腐蚀模型。通过深入研究这一领域，本项目旨在解决与交错相关的几个开放问题，并将推进对简单随机漫步的大偏差特性的理解。特别令人感兴趣的是在合适的时间尺度上环面上行走的空集的截断两点函数的行为。在确定精确的领先渐近行为的观察，无论是在亚和超临界制度，该项目旨在解决一个长期的核心问题，在这一领域。为此目的所获得的方法和结果很可能适用于涉及长期依赖性的其他情况。该项目属于EPSRC数学分析、概率和数学物理研究领域，特别是与随机结构、随机分析和临界现象有关。

Historically, it has been of great interest and practical relevance to study spreading phenomena of a fluid (water, pollutants,...) through a medium, and many Mathematical models have been introduced for this purpose. M. Sahimi (1994) assigns two possible categories to such models: diffusion processes and percolation processes. In the first, the assumption is that the source of randomness lies with the fluid, whereas in the second, the motion of the fluid is wholly determined by the structure of the medium, which itself is considered to be randomly generated. The point of view taken by percolation processes was initially studied and contextualised by S. R. Broadbent and J. M. Hammersley (1957), and has since led to further exploration of various random geometries such as the Bernoulli percolation model or Gaussian Free Field (GFF) excursion sets. The latter give rise to new and exciting universality classes, characterized by the presence of long-range correlations between local observables, which decay polynomially with the distance. All these models exhibit what is called a phase transition, presumably of second order and characterized by critical exponents, which takes place as the density changes from a subcritical phase, consisting of small connected islands only, to a supercritical phase comprising an infinite cluster. The goal of this research project is to further develop our mathematical understanding of long-range correlated percolation models. One such model on the integer lattice is called the random interlacements, and was first introduced by A.-S. Sznitman (2010) to deal with probabilistic covering and fragmentation problems attached to random walk traces. Such models were considered e.g. by M.J. Brummelhuis and H.J. Hillhorst (1991) as models of corrosion. By further delving into this area, this project aims to solve several open problems relating to the interlacements and will progress understanding of large deviation properties of the simple random walk. Of particular interest is the behaviour of the truncated two-point function for the vacant set of the walk on the torus at suitable timescales. In determining the precise leading asymptotic behaviour for this observable, both in sub- and supercritical regime, this project aims to tackle a long-lasting and central question in this area. It is likely that the methodology and results obtained for this purpose will be adaptable to other scenarios involving long-range dependence. This project falls within the EPSRC Mathematical Analysis, Probability and Mathematical Physics research areas, in particular relating to random structures, stochastic analysis and critical phenomena.