Minimum connectivity required for mean field to be a good approximation? In spin models, it is known that mean field becomes a better approximation as the connectivity increases. My question is: Is there an estimate for the threshold connectivity (as a function of the system size, $N$), such that for connectivites greater than the threshold, mean field is a good approximation?
The values I am mainly interested in are the free energy and the magnetization, so the mean field approximation should be good in the sense that these values are accurate. 
 A: Rigorous bounds comparing mean-field free energy and magnetization to their lattice counterparts, for various interaction ranges and dimensions, have been obtained in the following series of works (after many earlier works, see the references in these works):
arXiv:math/0207242
arXiv:math-ph/0501067
Commun. Math. Phys. 292 No 2 303–341 (2009)
However, note that the bounds they get are probably quite poor (this was not the main point of their work). They treat in details a few models (Potts, XY, etc.), but the approach is not too specific. It is however essential that the model be reflection positive.
Although, this is not your main interest, let me mention, just to complete the answer, the following work, in which rigorous bounds are obtained (using the lace expansion) allowing to derive mean-field values for critical exponents of Ising, percolation and SAW models in either sufficiently high dimensions, or with sufficiently spread-out interactions:
arXiv:0712.0312
There are many other rigorous works on this topic, of course, but this should get you started.
