It is easy to get lost in the math of the previous sections and lose sight of the intent of our work. Now is a good time to pause and review what we have accomplished by studying one-dimensional percolation.
First, we were able to find pc, the critical value of the probability at which an infinite cluster appears. This result is completely general to high dimensions, and lattice structures that are more complicated than square or cubic.
We also saw that certain quantities diverge as p approaches pc. In one dimension, the mean cluster size s and the correlation length diverges as 1/(pc-p) as p gets asymptotically close to pc. This type of divergence a power law, with an exponent of -1. It is quite common for quantities to diverge as a power law, though with different exponents.
The behavior of percolation is similar to the behavior of physical systems undergoing phase transitions. Quantites like S and ξ have physical counterparts in these transitions. For example, as the temperature (T of a gas decreases, the gas will undergo a phase transition into a liquid at a critical temperature Tc. As the T approaches Tc, the compressability of the gas divergers as a power law.
As a liquid turns into a gas, there is another example of critical behavior that is observable, sometimes to the bare eye. As the temperature of a liquid is increased towards Tc, some small groups of molecules fluctuate between liquid and gas. As the temperature asymptotically approaches Tc, the scale on which these fluctuations occur diverges. This is analogous to the correlation length in percolation theory. When the correlation length is large enough, the fluctuations are of sufficient scale to affect they way in which visible light scatters. Such an effect can also be seen in a binary mixutre of liquids. This phenomenon is critical opalesence and can be seen in this video.
Another example of such a phase transition is when between the paramagnetic and ferromagnetic phases of certain materials such as iron. If iron is hot, it will no longer be a ferromagnet (i.e. it will not stick on your fridge.) As the temperature decrease to a critical value known as the Curie temperature (Tc), the material will undergo spontaneous magnetization. In zero applied field, the magnetization will be non-zero when the temperature is below Tc and zero when it is above Tc. The magnetization vanishes as a power law as T approaches Tc from below. The magnetization, which is defined as the change in
The exact value of the exponent in the power laws are of great interest to physicists. This is because the exponent gives clues as to the underlying physics of the system. This is one reasone why physicists are interested in studying percolation theory: it can provide the tools to understand critical exponents in physical systems.