Renormalization is the phenomenon for which, once a finite number of parameters, which are the couplings with positive-mass dimension, are fixed, then it is possible to express any $n$-point correlation function of as a power series of these parameters.
The Renormalization Group approach allow to understand this fact as a consequence of being near a fixed point of the renormalizatio group flow (the free theory in the case of interest for high energy physics): this fixed point makes so that all negative mass-dimension couplings gets "eaten" by the flow. A possibly $(I+R)$-dimensional domain of the space of field theories gets squashed to a $R$-dimensional, where $R$ is the number of relevant parameters (mass dimension positive) and $I$ the number of irrelevant.
In statistical physics, a renormalization group fixed point is interpreted as a critical point, more precisely as a second order phase transitions. In fact it means that we have scale invariance, hence the correlation functions follow a power law and consequently second derivatives of the Free energy diverges.
Second order phase transitions are also associated to a spontaneous breaking of simmetry: this is the Landau theory of phase transitions which puts the emphasis on a parameter order.
My question is, from the field theoretical point of view, what is in general the interpretation of the parameter order? Can it always be understood in terms of the fields appearing in the lagrangian? In the space of theories, near the fixed point which corresponds to a free theory, there is a separation of theories in phases? If so, there is an interpretation, from the point of view of high energy physics rather than statistical physics, of this?