a) b - École Polytechnique de Montréal

lattice, such as square lattice, with a random network by randomly occupying sites (vertices) or
bonds (edges) with a statistically independent probability is imagined (Figure 2-16).
If it is examined as an infinite network corresponding to Kolmogorov`s zero-one law, for a
given P, the probability that an infinite cluster exists is either zero or one. There is a critical P
(since this probability is increasing) below which the probability is always 0 and above which
the probability is always 1. This critical amount Pc is called the percolation threshold, which if
we start out at P = 0 and randomly create connections, Pc is defined as a point at which a
spanning cluster first appears. It means that for P greater than Pc, a spanning cluster always
exists, although some isolated, non-spanning clusters can be present, and for P less than Pc, there
are only isolated clusters(Stauffer & Aharony, 1994a).
Figure 2-16. Detail of a bond percolation on the square lattice in two dimensions with
percolation probability p=51(Stauffer & Aharony, 1994b)
Pc may be calculated theoretically in some cases such as for a square lattice in two-dimensions
where the percolation threshold has a value of 0.5. For triangular and honeycomb bonds, (Pc)
will be respectively as follows:
⎛ π ⎞
Equation 2-38. Pc = 2Sin⎜
⎟
⎝18⎠
41