Unfortunately I'm not familiar with this field. Sorry.

Sure: take the Curie-Weiss mean field theory of ferromagnetism. It predicts a second order phase transition, between a paramagnetic, high temperature phase and a ferromagnetic, low temperature phase,  in any dimension. Fluctuations eliminate the low temperature phase in one dimension for the Ising approximation and, also, in two dimensions, for the Heisenberg ferromagnet (the planar restriction has a special phase transition in d=2, the Kosterlitz-Thouless transition, that's completely different from the transition described by the mean field approximation). In two dimensions and three dimensions, fluctuations don't eliminate the low temperature phase, in the Ising approximation but the physical properties are considerably modified.  

Disorder can affect-or not-the phase diagram, sometimes quite significantly. Mathematical results may be found here: https://www.ias.edu/people/faculty-and-emeriti/spencer, here: http://www.mathunion.org/ICM/ICM1986.2/Main/icm1986.2.1312.1318.ocr.pdf  

and here: http://www.princeton.edu/~aizenman/MA_pubs.html for instance.

For neural networks and disordered systems, a very nice reference is ``Field Theory, Disordered Systems and Simulations,  is http://www.worldscientific.com/worldscibooks/10.1142/1655, by Giorgio Parisi; cf., also, http://chimera.roma1.infn.it/GIORGIO/indexhome.htm and the home page of Marc Mézard, http://lptms.u-psud.fr/membres/mezard/lipub_recent.html and of Bernard Derrida, http://www.lps.ens.fr/~derrida/publi.html

While I have an undergrad degree in physics, this is sadly outside the areas I know. Haven't seen the Ising model in decades.... 

incidentally, ``finite-size'' here means something different from ``mean field corrections''. The corrections to mean field behavor are *finite-dimension'' corrections, due to the fact that each degree of freedom does not interact with all the others equally, in finite dimension.

``Finite size effects'' mean that one considers a finite number of degrees of freedom-and, in that case, the trnsition itself can be eliminated by quantum or thermal, or disorder fluctuations-only in the thermodynamic limit of an infinite number of degrees of freedom is the transiition sharp. Of course if the finite number is, in fact quite ``large'', the time the system takes to move from one phase to another will, also, be large and it may be large enough to be infinite for all practical purposes.

``Finite-size effects'' exist in the mean field approximation as well as in finite dimensional systems.

@Stam: thanks for your answers. I'm looking at the links you posted, it seems there is really a lot of information on the topic. Especially interesting is the part on neural networks, for what I'm currently working on.

@ Steven: no worries, I also forgot everything about the Ising model!

Have a nice Easter!