Quote by atyy
Does AS really need a fixed point? Could it live with, say, a limit cycle?

Yes, the WilsonKandoff renormalization group takes place on a block lattice with everything in units of the lattice spacing [tex]a = 1[/tex]. In lattice units the regularized integrals of perturbation theory have no divergences as [tex]a \rightarrow 0[/tex], because [tex]a[/tex] has disappeared.
However the problem of renormalization has been replaced by the problem of taking a continuum limit, with no [tex]a[/tex] where is the continuum limit [tex]a \rightarrow 0[/tex]. This problem is solved by the lattice correlation length, which roughly tells you how big correlations are in lattice units. If you fix the correlation length in physical units, then the lattice correlation length has to diverge as you approach the continuum, as lattice units are smaller and smaller compared to physical units.
So the continuum limit is associated with points with infinite lattice correlation length, which are fixed/critical points.