a) b - École Polytechnique de Montréal

⎛ π ⎞
Equation 2-39. Pc = 1−
2Sin⎜
⎟
⎝18⎠
Calculation of Pc for a square lattice was an open question for more than 20 years and was finally
resolved by Harry Kesten. For three-dimensional systems, Pc’s are unknown and cannot be
calculated for now, although the most interesting thing to do is not calculating the Pc.
The percolation threshold has a significant application in universality systems. For example,
Utracki et al.(Utracki, 1991) used the percolation threshold to predict the phase inversion point
based on emulsion theory as follows:
Equation 2-40.
η ⎛φm
φ ⎞
1 − 2 =
η ⎜
φm
φ ⎟
2 ⎝ − 1 ⎠
In this model, m and η refer to the volume fraction of the component at percolation threshold
and intrinsic viscosity, respectively.
The universality principle describes that the value of Pc is connected to the local structure of the
graph, while the behavior of clusters below, at and above Pc are invariants of the local structure,
and therefore, in some sense are more natural quantities to consider.
2.2.1 Universality Systems
Systems which tend to be chaotic often have a large number of interacting parts. They are called
universality systems. Systems display universality in a scaling limit, when a large number of
interacting parts come together. As a matter of fact, universality originated in the study of phase
transitions in a statistical science, especially in statistical mechanics. The transformation of a
thermodynamic system from one phase to another is a phase transition. When the properties of a
material change in a dramatic way, a phase transition happens. For example, when a magnet is
heated, its magnetism is lost. In this case, the phase transition is characterized by the
[ η ] φm
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