Is there a direct link between phase transitions in physics and Percolation Theory (and the critical points in both cases) in the sense that the mechanism in one of them could be a special or generalized case of the other?
One should also mention that there are several statistical mechanical models in which the phase transition can be understood as the percolation transition of a suitable (dependent) percolation model. The best known case is that of the Ising and Potts models, in which the transition occurs at the percolation transition of the associated FK percolation (random cluster) model (see, for example, here). Other examples are the (continuum) Widom-Rowlinson model (for example, here), the Ashkin-Teller model (for example here or here), etc. There are also examples for continuous spin systems, for example the XY model (see for example here).
Note that the relation between various graphical representations of this type and the phase transition in the original model provides powerful tools for their analysis; see these lecture notes for several remarkable recent examples).