This is a pretty naive question. I know of infrared conformal invariance in various second order phase transitions. I also know about the conformal U(1) boson at the XY critical point, an infinite order phase transition. I was always confused about the relationship between the order of the phase transition and the existence of a conformal fixed point. Do we expect CFT behavior at all critical points with order 2 and higher?

A model I'm particularly interested in right now is the Ising model on an infinite tree with, say, constant vertex degree. This model has a finite temperature, infinite order phase transition. I'd really like to know if this theory has a conformal fixed point. I'm not even sure what dimension it should live in. Probably somewhere between 1 and 2 like the tree itself...