The act of coarse-graining/scaling up (RG transformation) changes the theory that describes the system, specifically the theories parameters. If you consider in the space of all theories and iterate the coarse-graining, this induces a flow where each theory is mapped to a coarse-grained version. This flow may posess attractors, that is stable fixed points x*, meaning that when you apply the coarse-graining you get the same theory back.
And if f(x*)=x* then obviously f(f(x*))=x*, i.e. any repeated application will still yield the fixed point.
So you can scale up as much as you want—entering a fixed point really is a one way street, you can can check out any time you like but you can never leave!
The act of coarse-graining/scaling up (RG transformation) changes the theory that describes the system, specifically the theories parameters. If you consider in the space of all theories and iterate the coarse-graining, this induces a flow where each theory is mapped to a coarse-grained version. This flow may posess attractors, that is stable fixed points x*, meaning that when you apply the coarse-graining you get the same theory back.
And if f(x*)=x* then obviously f(f(x*))=x*, i.e. any repeated application will still yield the fixed point.
So you can scale up as much as you want—entering a fixed point really is a one way street, you can can check out any time you like but you can never leave!