It seems a lot of physical intuition in statistical mechanics, for example phase transitions, critical temperature, scaling hypothesis, renormalization group methods etc. should have a purely mathematical formulation; my question is: to what extent can this be done? Can we prove statements from a standard textbook (say Statistical mechanics by Huang) in a mathematically rigorous way?
A more specific example: it is well know that the 2D ising model with no external magnetic field has a 2nd order phase transition, can this be proven rigorously?