Renormalization

In this chapter we will introduce the powerful theoretical methods of renormalization. The fundamental idea is that at \(p=p:c\) , a rescaling of the system does not change the most important features. By a rescaling we typically mean a coarse-graining of the system, such as merging \(2 \times 2\) cells into a single cell. The rule we use to choose the occupation probability of the new, coarse-grained cell, \(p'\) , is a function of the probability p of the original lattice, \(p' = R(p)\) . In renormalization theory, we use properties of this mapping, \(R(p)\) , to deduce properties of the system such as critical exponents. In this chapter, you will be introduced to the fundamentals of renormalization theory in the context of percolation systems, in which the geometric nature of the remapping allow us to build intuition about renormalization as a concept. We will also apply the theory to different lattice structures and for one, two and three-dimensional systems.