Percolation on Soft Random Geometric Graphs

本项目的目的是研究软随机几何图(SRGG)上的渗流结果。SRGG是一个嵌入到度量空间中的随机连接模型，通常我们选择标准的欧几里德空间，更准确地说，我们分两步在我们的空间中建立一个图G。我们首先通过齐次泊松点过程来选择顶点。然后，我们根据点之间的距离将点随机连接在一起。我们可以用一个联系函数来描述联系的概率。一个这样的例子是这样一个函数，它给彼此距离小于1的所有点对规定概率p，给出其他的概率0。给出这样一个图，我们现在可以研究它的性质。这个研究项目旨在研究这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)。这一领域的一个经典结果是理解无限分量存在所需的条件--这种性质称为渗流。因此，渗流概率是特定节点连接到无限集群(通常为零的节点)的概率。更具体地说，存在关键参数，如果增加，则总是导致无限团簇，如果减少，则总是导致它们的消失。这为模型的研究创造了三个明确的机制：超临界、亚临界和临界。其他结果包括亚临界参数分量的指数衰减，超临界参数有限分量的指数衰减，以及更广泛地发展了类似于FKG不等式、BK不等式和Russo公式的工具等。我计划通过结合来自随机几何图研究的现有方法和来自Mathew Penrose、Geoffrey Grimmett、Ronald Meester和Rahul Roy等人的连续渗流来展示这些结果，这些方法是由文森特·塔塞斯、Hugo Dumil-Copin、Gabor Pete、Ioan Manolescu和许多其他人开发的较新的方法。这项工作的应用可能包括联网，在这种情况下，渗流属性对应于用户可以向系统中具有不可靠连接的另一用户发送消息的概率。在物理学中，渗流理论已经被用来理解量子自旋系统--因此，连续介质模拟有可能有类似的应用。

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.