|
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.〔R. Meester. ''Continuum percolation'', volume 119. Cambridge University Press, 1996. 〕〔M. Franceschetti and R. Meester. ''Random networks for communication: from statistical physics to information systems'', volume 24. Cambridge University Press, 2007. 〕 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,〔〔E. N. Gilbert. Random plane networks. ''Journal of the Society for Industrial \& Applied Mathematics'', 9(4):533–543, 1961. 〕 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.〔O. Dousse, F. Baccelli, and P. Thiran. Impact of interferences on connectivity in ad hoc networks. ''Networking, IEEE/ACM Transactions on'', 13(2):425–436, 2005. 〕〔O. Dousse, M. Franceschetti, N. Macris, R. Meester, and P. Thiran. Percolation in the signal to interference ratio graph. ''Journal of Applied Probability'', pages 552–562, 2006. 〕 In additions to this setting, continuum percolation has gained application in other disciplines including biology, geology, and physics, such as the study of porous material and semi-conductors, while becoming a subject of mathematical interest in its own right.〔I. Balberg. Recent developments in continuum percolation. ''Philosophical Magazine Part B'', 56(6):991–1003, 1987. 〕 ==Early history== In the early 1960s Edgar Gilbert 〔 proposed a mathematical model in wireless networks that gave rise to the field of continuum percolation theory, thus generalizing discrete percolation.〔 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,〔P. Hall. On continuum percolation. ''The Annals of Probability'', 13(4):1250–1266, 1985.〕 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「continuum percolation theory」の詳細全文を読む スポンサード リンク
|