Percolation on Soft Random Geometric Graphs

这个项目的目标是研究软随机几何图（SRGG）上的渗流结果。SRGG是一个嵌入在度量空间中的随机连接模型，通常我们选择标准的欧氏空间，更准确地说，我们在我们的空间中分两步建立一个图G。我们首先选择一个齐次泊松点过程的顶点。然后，我们根据点之间的距离将它们随机连接在一起。我们可以用连接函数来描述连接的概率。一个这样的例子是一个函数，它规定了所有点对的概率p，这些点之间的距离小于1，否则概率为0。给定这样一个图，我们现在可以研究它的性质。该研究项目旨在研究此类模型的大规模连通性，数学家称之为渗流理论（物理学家称之为统计物理学）。这个领域的一个经典结果是理解无限组分存在所需的条件-这种性质被称为渗透。渗透概率是特定节点连接到无限簇（通常是零节点）的概率。更具体地说，关键参数存在，如果增加总是导致无限的集群，如果减少总是导致他们的缺席。这创建了三个明确的制度下研究模型：超临界，亚临界和临界。其他结果包括显示指数衰减的组件的次临界参数，指数衰减的有限组件的超临界参数，更一般地开发类似的工具，以FKG不等式，BK不等式和Russo的公式，等等。我计划通过结合现有的方法来展示这些结果，这些方法来自随机几何图的研究和连续渗流，来自Mathew Penrose，Geoffrey Grimmett，罗纳德Meester和Rahul Roy等人的新方法，由Vincent Tabs，Hugo Dumil-Copin，Gabor Pete，Ioan Manolescu和许多其他人开发。这项工作的应用可能包括网络，在这种情况下，渗透特性对应于用户可以利用不可靠连接向系统中的另一用户发送消息的概率。在物理学中，渗透理论已经被用来理解量子自旋系统-因此连续模拟可能有类似的应用。

The goal of this project is to investigate percolation results on Soft Random Geometric Graphs (SRGGs). The SRGG is a random connection model embedded in some metric space, usually we choose the standard Euclidean space.More precisely, we build a graph G in our space in two steps. We first choose the vertices by a homogenous Poisson point process. We then randomly connect points together based on their distance from each other. We can describe the probability of connection by a connection function. One such example would be a function which prescribes probability p to all pairs of points which have distance less than 1 to each other, and probability 0 else.Given such a graph we can now investigate its properties. This research project aims to study large-scale connectivity properties of such models, what mathematicians call percolation theory (and physicists call statistical physics). One classical result in this field is understanding what conditions are required for infinite components to exist - this property is called percolation. The percolation probability is then the probability that a specific node is connected to the infinite cluster (usually the node at zero). More specifically, critical parameters exist, which if increased always lead to infinite clusters, and if decreased always lead to their absence. This creates three explicit regimes to study the model under: supercritical, subcritical and critical. Other results include showing exponential decay of components for subcritical parameters, exponential decay of finite components for supercritical parameters, and more generally developing analogous tools to the FKG inequality, the BK inequality and Russo's formula, amongst others. I plan to show these results by combining existing methods from the study of Random Geometric Graphs and continuum percolation from the likes of Mathew Penrose, Geoffrey Grimmett, Ronald Meester and Rahul Roy newer methods developed by Vincent Tassion, Hugo Dumil-Copin, Gabor Pete, Ioan Manolescu and many others.Applications of this work might include networking, in which case the percolation property corresponds to the probability that a user can send a message to another user in the system with unreliable connections. In physics percolation theory has been used to understand quantum spin systems - thus it is possible that the continuum analogue has similar applications.