# Singular behaviour in the Renormalization Group

We have stated that in general the RG transformations ${\displaystyle R_{\ell }}$ are analytic, so we might ask: where does the singular behaviour of a system near a critical point come from? Like what we have seen in Statistical mechanics and phase transitions this occurs in the thermodynamic limit, which in this case is obtained when we apply the RG transformation an infinite number of times.

In general, after the ${\displaystyle n}$-th iteration of the RG the coarse-graining length of the system will be ${\displaystyle \ell ^{n}}$ and the coupling constants ${\displaystyle [K^{(n)}]}$. As ${\displaystyle n}$ increases the "vector" of coupling constants describes a "trajectory" in the space of all the possible coupling constants, often called Hamiltonian space or theory space; we call RG flow the set of all the trajectories that start from different initial conditions, i.e. different initial Hamiltonians. In general, these trajectories can form strange attractors or complex limit cycles; however, it is almost always found that they are simply attracted towards or ejected from fixed points (cases where this doesn't occur are really exotic), so in the following we will assume that the RG flow only exhibits fixed points. The study of the properties of the RG flow near these fixed points is crucial, since as we will see it is that that will allow us to actually explain universality and predict the values of the critical exponents. We therefore proceed to study such points.