Connectivity functions for random walk percolation models

从历史上看，研究流体（水、污染物等）的扩散现象具有极大的兴趣和实际意义。通过一种媒介，许多数学模型已被引入为此目的。M. Sahimi（1994）将这类模型分为两类：扩散过程和渗透过程。在第一种情况下，假设随机性的来源在于流体，而在第二种情况下，流体的运动完全由介质的结构决定，介质本身被认为是随机产生的。渗透过程的观点最初是由S. R. Broadbent和J.M. Hammersley（1957），并从此导致了对各种随机几何的进一步探索，如伯努利渗流模型或高斯自由场（GFF）偏移集。后者产生了新的和令人兴奋的普适性类，其特点是存在的本地观测，随距离多项式衰减之间的长程相关性。 所有这些模型都表现出所谓的相变，大概是二阶的，其特征在于临界指数，这发生在密度从亚临界相，由小的连接的岛屿只，超临界相，包括一个无限的集群变化。 这个研究项目的目标是进一步发展我们对长程相关渗流模型的数学理解。整数格上的一个这样的模型被称为随机交织，并且首先由A. S. Sznitman（2010）处理随机游走迹的概率覆盖和碎片化问题。例如，M.J. Brummelhuis和H.J. Hillhorst（1991年）认为这种模型是腐蚀模型。 通过进一步深入研究这一领域，该项目旨在解决与交错相关的几个开放性问题，并将进一步理解简单随机游走的大偏差特性。特别令人感兴趣的是截断两点函数的空集上的环面在适当的时间尺度上的行走的行为。在确定精确的领先的渐近行为，这可观察到的，无论是在亚和超临界政权，该项目的目的是解决一个长期存在的和中心的问题，在这方面。为此目的采用的方法和取得的结果很可能适用于涉及长期依赖性的其他设想。 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的福尔斯。

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.