Continuum percolation theory

[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.

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.

A major focus of percolation theory is establishing the conditions when giant components exist in models, which has parallels with the study of random networks.

The conditions of giant component criticality naturally depend on parameters of the model such as the density of the underlying point process.

For example, in a system of randomly oriented homogeneous rectangles of length l, width w and aspect ratio r = ⁠l/w⁠, the average excluded area is given by:[11] In a system of identical ellipses with semi-axes a and b and ratio r = ⁠a/b⁠, and perimeter C, the average excluded areas is given by:[12] The excluded area theory states that the critical number density (percolation threshold) Nc of a system is inversely proportional to the average excluded area Ar: It has been shown via Monte-Carlo simulations that percolation threshold in both homogeneous and heterogeneous systems of rectangles or ellipses is dominated by the average excluded areas and can be approximated fairly well by the linear relation with a proportionality constant in the range 3.1–3.5.

Using the tools of percolation theory, a blinking Boolean Poisson model has been analyzed to study the latency and connectivity effects of such a simple power scheme.