The mathematical model of the percolation was introduced by Hammersley in 1957. It is interested in the characteristics of the random mediums, more precisely with the whole of tops connected in a random Graphe.

Informellement, let us imagine that one places water at the top of a spongy stone. If there are enough small channels, it is then possible that there is a way of the center of the stone towards outside. This model makes it possible to answer this kind of question.

This theory applies to the Science of the materials, in the field of the Percolation.

Description of the basic model

We consider the D-dimensional network $Z^d$. The edges are the couples of items remotely Euclidean 1. We fix then a parameter $p$ ranging between 0 and 1. Each edge is then open with closed probability $p$ and with probability $1-p$, this independently from/to each other. The open denomination means that the edge is kept, whereas closed means that it is removed. One notes $P\_p$ measurement obtained.

One is interested in the existence of an infinite way in the random graph thus obtained. An essential quantity is the probability of percolation: $\backslash \; theta\; (p)\; =P\_p\; (O\; \backslash \; rightarrow\; \backslash \; infty)\; \backslash ,\; \backslash !$. By arguments of coupling, it is easy to show that $\backslash \; theta$ is an increasing function of $p$.

There exists a critical point $p\_c$ such as $\backslash \; theta\; (p)$ is null if

The subcritical mode

In this mode, there is no infinite way in the graph. The related components (also called clusters) finished are generally small size. More precisely probability than the cluster containing the point $x$ a size has which exceeds $n$ decrease exponentially quickly with $n$. In particular, the intermediate size of a cluster is finished.

The mode criticizes $pp\_c$

This mode is still badly known (except notable for dimension 2). One conjectures that $\backslash \; theta\; (p\_c)\; =0$, i.e. there is no percolation at the critical point, but this is for the moment shown only in dimension two or great dimension $d>18$. In particular, the case of the dimension three, whose physical relevance is obvious, remains not proven.

The supercritical mode $p>p\_c$

In the supercritical phase, there is a single infinite component of connected points. However, the finished clusters are generally small size. The infinite cluster meets all space; more precisely the proportion of points of a box of size $n$ which belong to the infinite cluster tightens towards $\backslash \; theta\; (p\_c)\; >0$ when $n$ tends towards the infinite one. It is also known that the infinite cluster is very rough: the proportion of the points of a box of size $n$ which are at the border of the infinite cluster among the totality of the points of the infinite cluster which are in this limps tends towards $1-p$ when $n$ tends towards the infinite one.

Other models

• the directed Percolation which with bonds with the Process of contact
• the Percolation FK which makes it possible to connect the percolation to the Modèle of Ising and with the Modèle of Potts.
• the Percolation of first passage

Books of references

• Percolation of G.R. Grimmett at Springer
• Percolation Theory for Mathematicians of Harry Kesten at Birkhaüser

• Introduction to Percolation Theory, Taylor and Francis ED. London, 1985

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