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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?

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  • $\begingroup$ Percolation is a phenomenon characterised by the fact that a "typical" length blowsup. Note that the length can be thought as a correlation between local observables, but this is not necessary. Under a rescaling of the system - in this case the rescaling is given by zooming out but in other case it could be zooming in or integrating away small or fast energies - this length has to scale. However if it is infinity, the rescaling leave it unaffected, hence the system looks the same at every ditance. This happens anytime you have a 2nd order phase transition. $\endgroup$ – giulio bullsaver Oct 18 '17 at 14:33
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Yes. Percolation has a critical point where a continuous phase transition takes place: the percolation transition is therefore a particular case of phase transition.

You can read more about it in the Wikipedia pages on percolation and its critical exponents.

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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).

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