**The Renormalization Group trick:**

Suppose you have some object $v_0$ and you want to understand a feature $Z(v_0)$ of that object. First identify $v_0$ as some element of a set $E$ of similar objects. Suppose one can extend the definition of $Z$ to all objects $v\in E$. If $Z(v_0)$ is too difficult to address directly, the renormalization group approach consists in finding a transformation $RG:E\rightarrow E$ which satisfies $\forall v\in E, Z(RG(v))=Z(v)$, namely, which preserves the feature of interest. If one is lucky, after infinite iteration $RG^n(v_0)$ will converge to a fixed point $v_{\ast}$ of $RG$ where $Z(v_{\ast})$ is easy to compute.

Example 1: (due to Landen and Gauss)

Let $E=(0,\infty)\times(0,\infty)$ and for $v=(a,b)\in E$ suppose the "feature of interest" is the value of the integral
$$
Z(v)=\int_{0}^{\frac{\pi}{2}}\frac{d\theta}{\sqrt{a^2\cos^2\theta+b^2\sin^2\theta}}\ .
$$
A good transformation one can use is $RG(a,b):=\left(\frac{a+b}{2},\sqrt{ab}\right)$.

Example 2: $E$ is the set of probability laws of real-valued random variables say $X$ which are centered and with variance equal to $1$. The feature of interest is the limit law of $\frac{X_1+\cdots+ X_n}{\sqrt{n}}$ when $n\rightarrow\infty$. Here the $X_i$ are independent copies of the original random variable $X$.

A good transformation here is $RG({\rm law\ of\ }X):={\rm law\ of\ }\frac{X_1+X_2}{\sqrt{2}}$.

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