How renormalization allows to describe critical point behaviour using the critical fixed point? As in the title, I am trying to understand how the critical fixed point (CFP) can be used to derive the thermodynamic singular behavior of the physical critical point (PCP). The context I have in mind is that of the 2d Ising model in real space renormalization group (RG).
Here I just present the basic notions, since maybe my misunderstanding is give by some false idea I have.
I imagine we have a RG transform that

*

*leads to analytic recursion relations

*leaves invariant the total free energy

*has a CFP and its associated stable manifold (the critical manifold) whose dimension is given by the number of irrelevant variables relative to that CFP.

*the PCP is the intersection among the critical manifold and the curve of NN (nearest neighbour) Ising model hamiltonians.

*the hamiltonian at the PCP point under the RG transform follows a critical trajectory that ends only asymptotically at the CFP

Now here is what I think should be an argument to show that the CFP entails the PCP:

*

*scaling laws and critical exponents are asymptotic properties of the PCP, i.e the are valid asymptotically in a neighborhood of the PCP, so I should study how the system behave for values $t=T-T_C\ll1$ and $h\ll1$

*the analytic property of the RG transform allows us to expand in a Taylor series the recursion relations (or the RG differential equation) anywhere. Though the linearization procedure is accurate only in a neighborhood of the CFP, since this is a fixed point of the transformation. This justify our curiosity for the CFP

*I take as starting point $(t,h)$ near the PCP, and apply the RG transform. Since the transformation is analytic we expect that the renormalized point $(t',h')$ will not have a much greater distance from the relative renormalized critical point, respect the distance among the starting point and the PCP.

*We hope that the assumption in 3. holds till we arrive in a neighborhood of the CFP. That is we hope that our RG transform will get us a trajectory that is almost "parallel" to the critical trajectory. To justify this I guess is not trivial since we can't take the linearization of the RG to be accurate in describing the evolution of the starting point in parameter space, the full RG transformation is needed.

*if 4. is true then we know that points matched by the RG transformation describe the same thermodynamics, so the thermodynamic behaviour derived in the neighborhood of the CFP entails the one of the PCP neighborhood.

*from this the whole linearization procedure around the CFP can be used to derive the scaling form of the free energy and the whole critical exponents machinery.

Now I am far from being sure that this is correct.
Usual exposures I have seen in books don't really explain this and maybe avoid the problem by not keeping a net distinction between the CFP anf PCP. Usually, for what I have seen, their explanation consists just in the linearization of the RG transformation near the CFP and how from this we can get the free energy scaling laws and all the critical exponents. This is done using simply the invariance of the total free energy $F$ and hence the following equation for the free energy per particle $f$, $f[t,h]=b^{-2n}f[b^{ny_t}t, b^{ny_h}h]$. Though I think that to make sense, the starting point given by $t$ and $h$, in their context, should be a point near the CFP and not the PCP, otherwise the linearization isn't right.
Now the question per se is in the title how critical fixed point entails the singular behaviour of the physical critical point. I explained what I think I know, though I am far from sure this is correct. Besides, since my knowledge is only superficial, I feel like that even if it happens that what I said is correct ( I have some doubts), there is a lot of space o maneuver to explain the concept in a more rigorous, complete way. Hence Avoiding this to become a check my reasoning question.
I guess it is hard to be quantitative since I imagine that one should take a specific instance of RG transform, and try to prove 3. and 4.
 A: The description by the OP of a typical RG flow close to the critical point (called there PCP) is pretty much correct.
And, unfortunately, it is true that the distinction between a full trajectory and a linearized trajectory is usually put under the rug (it is even worse so in usual discussions of the epsilon expansion).
There is only one thing that is not quite right. The OP should have distinguished between the distance from the physical fixed point (call it $\tau=1-T/T_c$) and the distance from the (critical) fixed point along its relevant direction (call it $t$).
The distinction is important, because in systems where there are multiple microscopic parameters (say, couplings between nearest-neighbors, second nearest-neighbors, etc), typically all of them can be tuned to put the system at (or away from) criticality. Hence, they all have an overlap with the relevant direction $t$, even though they are not related to the temperature.
So the correct way to recover the scaling behavior is as follows (ignoring $h$).

*

*If $\tau$ is small enough (i.e. the system is close to the critical manifold), the flow will go towards the fixed point (even if we start very far from the fixed point, by definition of the critical manifold).

*This finite $\tau$ implies a finite $t$ in terms of the linearized flow, i.e. $t$ is a function of $\tau$, that vanishes with $\tau$ (and has the same sign). Because the flow is analytic, $t$ is a nice function of $\tau$. Unless there are some specific symmetries or some additional fine-tuning, we generically have $t\propto \tau$ for $\tau$ small enough with some non-universal constant of proportionality.

*The scaling of the free energy as $|t|^{1/\nu}$ implies a scaling $|\tau|^{1/\nu}$ in terms of the physical parameter $\tau$.

Non linear terms in $\tau$ appearing in $t(\tau)$ will only contribute to (analytical) correction to scaling.
