Water boiling and 3D Ising model I've been told for a long time that water boiling near critical temperature and the 3D Ising model near critical temperature are described by the same laws, and give a CFT. This is usually mentioned after the RG flow is introduced, and I understand that both Lagrangians are meant to be fixed points.
Where can I find an explicit proof that the laws defining water boiling near critical temperature and the 3D Ising model are the same? I assume this means showing they have the same Lagrangian.
 A: If you understand the key content of the Renormalization Group theory RGT), there is no need they should have the same Lagrangian, (well, in a context of Statistical Mechanics it would be more appropriate to speak in term of Hamiltonian, but this is a side remark).
The explanation of the existence of universality classes provided by RGT is based on the fact that fixed points in an abstract space, where each point corresponds to a different system at different physical conditions, do control the critical behavior of all the systems in the same basin of attraction.
Therefore, lattice systems with the same symmetry and spatial dimensionality but with different interactions do have the same critical exponents if the controlling critical point is the same.
On the one hand, a lattice gas model captures most of the relevant physics of a fluid system, whatever is the atomic Hamiltonian and this is is an argument in favor of the same critical point of the Ising model. On the other hand, more specialized implementation of RGT  for fluid systems have been proposed in the literature (for example the implementation based on correlation functions by Reatto, Parola and coworkers: Parola, A., and L. Reatto. "Hierarchical reference theory of fluids and the critical point." Physical Review A 31.5 (1985): 3309.) arriving again to the conclusion that the universality class of liquid-gas transition is the same of an Ising model at the same dimensionality.
