In mathematics and probability theory, continuum percolation theory is a branch of mathematics that extends discrete percolation theory to continuous space (often Euclidean space). More specifically, the underlying points of discrete percolation form types of lattices whereas the underlying points of continuum percolation are often randomly positioned in some continuous space and form a type of point process. For each point, a random shape is frequently placed on it and the shapes overlap each with other to form clumps or components. As in discrete percolation, a common research focus of continuum percolation is studying the conditions of occurrence for infinite or giant components.[1] [2] Other shared concepts and analysis techniques exist in these two types of percolation theory as well as the study of random graphs and random geometric graphs.
Continuum percolation arose from an early mathematical model for wireless networks,[2] [3] which, with the rise of several wireless network technologies in recent years, has been generalized and studied in order to determine the theoretical bounds of information capacity and performance in wireless networks.[4] [5] In addition to this setting, continuum percolation has gained application in other disciplines including biology, geology, and physics, such as the study of porous material and semiconductors, while becoming a subject of mathematical interest in its own right.[6]
In the early 1960s Edgar Gilbert[3] proposed a mathematical model in wireless networks that gave rise to the field of continuum percolation theory, thus generalizing discrete percolation.[2] The underlying points of this model, sometimes known as the Gilbert disk model, were scattered uniformly in the infinite plane according to a homogeneous Poisson process. Gilbert, who had noticed similarities between discrete and continuum percolation,[7] then used concepts and techniques from the probability subject of branching processes to show that a threshold value existed for the infinite or "giant" component.
The exact names, terminology, and definitions of these models may vary slightly depending on the source, which is also reflected in the use of point process notation.
A number of well-studied models exist in continuum percolation, which are often based on homogeneous Poisson point processes.
Consider a collection of points