Behavior in renormalization group flow that reaches critical point

Question

1. First question. Does correlation length in renormalization group flow has to be infinite when it eventually reaches critical point?

2. Second question. Why does renormalization group flow keep partition function invariant? In some sense, I don't see why probability in canonical ensemble has to be kept same, as after renormalization group flow, things may look different.

3. Third question. If a fixed point is not a critical point, does it have to have zero correlation length? Or can it still have infinite correlation length? Is it matter of definition?

4. Fourth question. Can there ever be a renormalization group that starts from a fixed point? Or is it impossible? Would there be a difference between critical points and non-critical fixed points?

5. Fifth question. Does correlation length has to decrease for any renormalization group flow?

Answer 1

Q1: One should not confuse critical point and RG fixed point. These are two different concepts. One expects that in the set $T$ of all theories, there is a hypersurface $\Sigma$ called the critical surface. When considering a specific model like the Ising model, that amounts to looking at say a curve $C\subset T$. For example a point $\gamma(\beta)$ on this curve would be parametrized by the inverse temperature. The critical point $\gamma(\beta_c)$ is the intersection $C\cap \Sigma$. The RG fixed point $P_{\ast}$ is a point on $\Sigma$ which satisfies $R(P_{\ast})=P_{\ast}$ where $R:T\rightarrow T$ is the renormalization group transformation. The relation between the two concepts is that $\lim_{n\rightarrow \infty} R^n(\gamma(\beta_c))=P_{\ast}$. Typically all the points on $\Sigma$ have infinite correlation length (you can even take this to the definition of $\Sigma$).

Q2: The RG leaves the partition function invariant by design. Wilson invented it do exactly that.

Q3: As explained in Q1, the (nontrivial) fixed point $P_{\ast}$ will have infinite correlation length like everybody else on $\Sigma$ and in particular the critical point too. That being said, there are also (very) trivial fixed points like the infinite temperature fixed point which are not on $\Sigma$. For those the correlation length is $0$. For more on this see my MO answer: https://mathoverflow.net/questions/270088/why-is-conformal-invariance-only-possible-for-massless-theories/270104#270104

Q4: Yes. An RG trajectory (for a discrete RG transformation $R$ rather than flow) is a bi-infinite sequence of forward and backward iterates $(R^n(P))_{n\in\mathbb{Z}}$ of some point $P$, inside the space $T$. Granting the existence of the limits $P_{\pm}=\lim_{n\rightarrow \pm\infty}R^n(P)$, these have to be RG fixed points. So one can say that the trajectory "starts" at $P_{-\infty}$ and "ends" at $P_{\infty}$. Careful though: if one is exactly at $P_{-\infty}$ one stays there forever, i.e., one cannot "start" to go anywhere else. For more on this see: What is the Wilsonian definition of renormalizability?

Q5: Yes. For instance if $R$ corresponds to zooming out by a factor of $2$, then the new correlation length (after applying the RG transformation $R$) is exactly the old one divided by $2$.