I recently came across this overview which discusses some results in the theory of critical phenomena. It is already quite old and I would like to know if there are other (more recent) overviews in the same lines of this one.
I am particularly interested (but not restricted to) in the following question. In the physics literature, it is often said that correlation functions of lattice spin systems satisfy a decay: $$\langle \varphi(x)\varphi(y)\rangle \sim f(|x-y|)e^{-|x-y|/\xi}$$ for some sub-exponential function $f$. Here, $\xi$ is the correlation length. The correlation length is supposed to satisfy $\xi \to \infty$ at criticality, so that at the critical point the above correlation has a decay of the form: $$\langle \varphi(x)\varphi(y)\rangle \sim \frac{1}{|x-y|^{\alpha}}$$ for some exponent $\alpha$.
I would like to better understand to which degree this is true. More precisely, to which models does this reasoning apply and what rigorous results we already have in this direction.
I am not looking for a paper which proves general results in a 100 pages. Instead, I am looking for an overview which gives the reader some definitions and the basic ideas and points out references where these results were discussed or proved.
