First and second order phase transitions: if there is a
finite discontinuity in one or more of the first derivatives of the appropriate
thernodynamic potential the transition is termed first order.
For a magnetic phase transition, the free energy (F=U-TS)
is the appropriate potential (F= kTln Z, U - internal energy, T - temperature, S - entropy)
with a discontinuity in the magnetization showing it is first order.
If the first derivatives are continuous but higher derivatives are not it is a higher order transition. Here the magnetization would be continuous.
Critical exponents describe the nature of the divergence in a second order transition. There are accepted names for the exponents of the
common quantities in the Ising model which are:
beta for the magnetization - not be confused with beta =1/kT in its Hamiltonian
gamma for the susceptibility
alpha for the specific heat
Upper and lower critical dimensions.
In mean field theory (alpha, beta, gamma) take the values
for the Ising model.
In one dimension, the exact solution of the Ising model does not
have a finite temperature transition.
In the exact solution of the 2D Ising model the exponents
take the values (0 (logarithm),1/8,7/4)
In 3d the exponents also differ from the mean field ones and
only in 4D do they take the mean field values (albeit with logarithmic
The lowest dimension which has a finite temperature transition is called
the lower critical dimension and the dimension at which exponents
become mean-field like is called the upper critical dimension. For the
Ising model it is D=4.
This issues are rather deep, and a full explanation requires scaling theory
and Renormalization Group theory. However the facts above
are all thats needed to understand the topic of percolation.
After all percolation is so easy you could explain it to your wife...........