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As far as I understand, the RG theory, or functional RG theory is a mathematical tool for moving in the "scale dimension". The tool can be used for calculation of Feigenbaums constant (e.g. mentioned here). Can the theory be given a simple example of how to move one "step" in the "scale dimension" ?

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    $\begingroup$ Consider the spin-1/2 Ising model on a triangular lattice (see, e.g. Goldenfeld). $\endgroup$ – Steve Huntsman Apr 6 '11 at 13:22
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The simplest and earliest example I know regarding the renormalization group idea is the following.

Suppose we want to study some feature $\mathcal{Z}(\vec{V})$ of some object $\vec{V}$ which is in a set $\mathcal{E}$ of similar objects. Suppose that unfortunately this question is too hard. What can one do?

The renormalization group philosophy is try to find a "simplifying" transformation $RG:\mathcal{E}\rightarrow\mathcal{E}$, such that $\mathcal{Z}(RG(\vec{V}))= \mathcal{Z}(\vec{V})$, and $\lim_{n\rightarrow \infty} RG^n(\vec{V})=\vec{V}_{\ast}$ with $\mathcal{Z}(\vec{V}_{\ast})$ easy to understand.

Example (Landen-Gauss, late 1700's):

Let $\vec{V}=(a,b)\in\mathcal{E}=(0,\infty)^2$ and consider $$ \mathcal{Z}(\vec{V})=\int_{0}^{\frac{\pi}{2}} \frac{d\theta}{\sqrt{a^2 \cos^2\theta+b^2\sin^2\theta}}\ . $$

A good choice of renormalization transformation here is $RG(a,b)=\left(\frac{a+b}{2},\sqrt{ab}\right)$, as discovered by Gauss.

A recent example now.

Example (Kadanof-Wilson, late 1960's early 1970's):

Take $\mathcal{E}$ to be the set of Borel probability measures on $\mathbb{R}^{\mathbb{Z}^d}$. Let $\mathcal{Z}(\vec{V})$ be equal to $1$ if the two-point function decays exponentially and $0$ otherwise.

Then define $RG$ as the transformation which gives the law of the block-spinned/coarse-grained field as a function of the law of the original field.

Note that the feature $\mathcal{Z}$ that one would like to preserve can be defined a bit more loosely. One could, e.g., "define" it as the long-distance behavior of the random spin field with probability law $\vec{V}$ (or in physics jargon: the low energy effective theory). Also in the dynamical systems context, it could be the chaotic behavior or not of a map $\vec{V}$. Then $RG$ could be a doubling transformation, i.e., composition of the map with itself together with some rescalings and reversals of orientation.

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When you look at the Mandelbrot set M, you can observe infinitely many "Baby Mandelbrot" copies of the set within M and so on. Renormalization allows you precisely to move from one step (the Baby copy) to the next one. For more details, see for example the book by McMullen "Complex dynamics and renormalization", or that article by M. Lyubich: https://web.archive.org/web/20150118224241/http://smf4.emath.fr/Publications/Gazette/2007/113/smf_gazette_113_45-50.pdf

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