I am currently trying to get my head round RG in the context of statistical mechanics and am not succeeding! I would be grateful for any help. I have a specific question, but any clarification of RG in general would be useful. Here is my understanding of the main ideas: 1. Define microscopic system with some Hamiltonian H. 2. Coarse grain this system in some (arbitrary) fashion, which gives rise to a system with a new H in terms of the coarse grained variables. 3. Now we use some input from experiment: near the critical point the system appears to be self similar at all scales, so we impose the requirement that H is of the same form in the new system so that the coarse grained system behaves as it should; the new H has some different set of constants in general given by a transformation. 4. We can then go on to find critical exponents and the critical coupling. The problem comes when I read or hear statements such as in Leo Kadanoff's book where he says that "each phase of the system can be described by a special set of couplings, K*, which are invariant under the RG transform". The procedure outlined above (if i've got it right!) only applies around the critical point from step 3 onwards, as elsewhere the system certainly doesn't look the same on all scales. Why does the recursion relation we previously obtained have any meaning away from the critical point? Thanks in advance!