Perspective on the renormalization group

Question

From reading the renormalization group description from high energy theory texts like peskin & schroeder, one may be tempted to think it has to do with regulating infinities.

However, my impression of the renormalization group once you rotate to euclidean time is that it is a mapping transformation of probability weights under a change of variables.

With this perspective which of those statements are true?

1. The gaussian fixed point is just a statement of the central limit theorem (CLT). Under weak enough correlations, CLT state one "flows" to a gaussian fixed point.
2. "Interacting" fixed points is then more interesting limiting distributions, aka generalization of the CLT. The anomalous critical exponents are just corrections to CLT like behavior. They are therefore fractal dimensions. Studying CFT is the study of fractal dimensions arising from interactions.
3. With this perspective, Brownian motion is a key example of a gaussian fixed point (under rescaling, the standard deviation scales as a root, hence critical exponent is $\frac{1}{2}$.

Answer 1

The high-energy physics point of view of the RG ("putting infinities under the rug") is now quite dated, but unfortunately is usually still the first version of RG that one enconters.

A more modern implementation a la Wilson (only 50 years old now...) can indeed be interpreted as a transformation of probability weights under coarse-graining (in HEP, this would correspond to effective theories).

This point of view was already quite present in the `70s, for instance in the papers by Jona-Lasino (which tend to the more mathematical side). One possible reference is "Critical point behaviour and probability theory" by Cassandro and Jona-Lasinio (DOI:10.1080/00018737800101504).

Indeed, from this point of view, the gaussian fixed point is just the statement that the CLT is valid asymptotically, while non-trivial fixed points correspond to breaking of the standard CLT (corresponding to different asymptotic PDF).