Percolation

Percolation theory is a theory which is used to describe varying numbers of connections in a random network. Take for example an <math>L \times L</math> array of holes on a substrate. One deposits small dots of metal, which can only sit in the holes on the substrate. Conduction can occur between the metal dots, because when two adjacent holes are filled with a metal dot, they just barely touch each other, thus allowing conduction to occur between them. Groups of touching metal dots are called clusters. A cluster which extends from one end of the array to the other is called a "spanning cluster". When you first begin depositing metal dots, there can be no conduction. There cannot be any conduction until at least L dots have been deposited; however, the statistical probability of the <math>L</math> dots aligning themselves to form a spanning cluster. Many more metal dots will need to be deposited before the probability of a spanning cluster becomes significant. At some point there will be an exponential increase in the conduction. This critical point is called the percolation threshold, below which no conduction can occur.