Functional equations

Functional equations show up a lot in results of the renormalization group. For example, $f(x^3)=f(x)/3$ represents the the analytic flow equation of the 1D Ising model under decimation. Another example is the scaling function of correlations of scaling fields:

\[ G(r/b) = b^{2x}G(r) \]

I have found the solutions of these functional equations mysterious, so I present their solutions below. The general scheme is that of perturbation theory.

Solving $ G(r/b) = b^{2x}G(r) $

Assume $b=1+\epsilon$ where $\epsilon \ll 1$, so that this is a perturbative rescaling of $r$. In this case we can expand the small parameter $\epsilon$:

is the left-hand side of the functional equation. The right-hand side is

To first order in $\epsilon$ our functional equation reads

$G(r)$ cancels on both sides, and then $\epsilon$ cancels! The result is a seperable differential equation

which has the solution

For some constant $C$. This is the classic ‘scaling dimension’ result.

Solving $ n f(x^n) = f(x) $

In the context of block-spin Renormalization this equation clearly intends for $n\in\mathbb{Z}$, but we can assume $n$ is any nonzero number. This equation clearly has no unique solution for $n=1$. So to proceed with our perturbation analysis let’s assume $n=1+\epsilon$. In this case the trickiest part is expanding $x^{1+\epsilon}$. This can be done by writing it as

Expanding the functional equation and simplifying as above yields another seperable differential equation

which has the solution

For some positive constant $C$. This is the result that allows the RG of the 1D Ising model to predict exponentially decreasing correlations at all finite temperatures.