Geometric interpretation of Renormalization Group (RG) flow in statistical mechanics? Is there a way of visualising RG flow in statistical mechanics in a geometric way? I am very new to this subject so it is probably a naive question, but it seems that as we are looking for fixed points of the transformation it might be helpful to do so, like in hamiltonian dynamics. I don't find the idea of renormalization particularly intuitive, so this would make it easier to understand.
(Sorry if this is a stupid question, but googling "geometric interpretation of RG flow" only gave results in QFT that are far beyond my level.)
 A: Ok. Renormalization group is a framework to stud statistical quantities in the continum limit. The continum limit in statistical mechanics is more subtle than usual because fluctuations occurs in all scales. A statistical quantity can fluctuate, and there is no smooth equation holding this quantities in a particular scale. What can happens is that fluctuations at sufficiently small scale of lenght can interfere in the long scale if there is  some non-linearity on the hamiltonian.
In the continum limit the partition function is 
$$
Z=\int \mathcal {D}\phi e^{-H[\phi]}=\int\prod_{k=0}^{\infty}d\phi_k e^{-H(\phi_1,...)}
$$
A sum over configurations of the field $\phi$. If there is a non-linearity in H, i.e. a non-bilinear term as $\phi_i\phi_j\phi_k$, the modes at diferent scales can couple. If the hamiltonian is linear, i.e. there is only terms as $A_{ij}\phi_i\phi_j$, you can diagonalize $A_ {ij}$ and obtain separate modes.
The approach of the RG is to apply the continum limit by cutting of the higer modes at first:
$$
\prod_{k=0}^{\infty}\rightarrow\prod_{k=0}^{\Lambda}
$$ 
And then seeing how the hamiltonian changes with $\Lambda$:
$$
e^{H_{\Lambda}}=\int\prod_{k=\Lambda}^{\Lambda'}d\phi_k e^{-H_{\Lambda'}}
$$
This is the right scale transformation of a partition function. That is, when you don't want to work with some mode, you should integrate over then, not only discard. This is so because he could interfere in our modes of interest.
Note that if the hamiltonian is bilinear you can separate the modes into a product of exponentials. The partition function would factorize and the integration over some mode don't affect the others.
Imagine a membrane in a cylinder with Dirichlet boundary condition. The hamiltonian is:
$$
H=\int rdrd\theta\, \phi(wave\,eq.)\phi
$$
You solve the wave equation with the proper boundaries conditions and then you have the modes of your problem. Now, you may try to insert some non-lineraity like a anharmonic term and couples this modes.
A: The concept of flow is emphasized in Wilson's approach (see the review article chapter 8-11, and in particular figure 5). One considers the set of infinite dimension space $\mathbb{H}=\{(K_0,K_1,K_2,\ldots)\}$ of couplings in Hamiltonian (with degrees of freedom $\{s_i\}$)
$$\mathcal{H}=K_0+K_1\sum s_i + K_2\sum s_is_j + K_3 \sum s_is_js_k+\cdots$$
The renormalization group defines a transformation $\mathbf{R}_b$ (with rescaling factor $b$) in the space of Hamiltonians $\mathbb{H}$, for $\mathbf{H}',\mathbf{H}\in\mathbb{H}$, as
$$\mathbf{H}'=\mathbf{R}_b(\mathbf{H})$$
In condensed matter physics one focuses on some specific form of $\mathcal{H}$, e.g., the Ising model with just two couplings, (reduced) temperature $t$ and external magnetic field $h$. (Page 665 column two of the article:) An important feature of Wilson’s approach, however, is to regard any such ‘‘physical Hamiltonian’’ as merely specifying a subspace (spanned, say, by ‘‘coordinates’’ t and h) in a very large space of possible (reduced) Hamiltonians, $\mathcal{H}$.
The subspace, e.g., physical manifold $\mathbb{H}_0$, will flow in $\mathbb{H}$ under successful renormalization transformation, in which the critical point of $\mathbb{H}_0$ will transform in to a fixed point $\mathcal{H}^*$. We may, however, start with a different initial physical manifold $\mathbb{H}_1$ such that under renormalization transformation, the critical point of $\mathbb{H}_1$ transform into the same fixed point $\mathcal{H}^*$. This is the idea of universality.
