Thermodynamics - unlike the phase transitions in spin models such as
the Ising model, which have the underpinnings of thermodynamics, percolation
is an order-disorder transition which does not come with this baggage.
Actually, in some ways because of recent work by mathematicians, the basis for
percolation is somewhat better because there is an exact
proof of hyperscaling in two dimensions.
However there are analogies with magnetic transitions - formalized
either via a dilute Ising model or thru the Potts mapping
leading to an order parameter that we call P(p), the
``Percolation Probability'' which has the critical exponent beta
like the magnetization. It is zero at Pc
and normalised to one at p=1.
(Mirror image of a magnetization graph.) The ``Mean Cluster Size'', S(p),
is the analog of the susceptibility which diverges with critical exponent gamma.
In site (bond) percolation each site (bond) is randomly occupied with
probability p and is empty with probability q=1-p. Our goal is
to determine the threshold value pc at which an infinite
(spanning) connected cluster of occupied nearest-neighbor sites (bonds)
occurs, and to describe the nature of the phase transition into the
The percolation order parameter P(p) is
the probability that an occupied site (bond) belongs to the infinite cluster.
quantity P(p) is zero for p < pc and behaves