For low–dimensional systems, (i.e. 2D and, to a certain extent, 1D) it is proved that mean–field theory can provide an asymptotic guideline to the phase structure of actual systems. In particular, for attractive pair interactions that are sufficiently “spead out” according to an exponential (Yukawa) potential it is shown that the energy, free energy and, in particular, the block magnetization (as defined on scales that are large compared with the lattice spacing but small compared to the range of the interaction) will only take on values near to those predicted by the associated mean–field theory. While this applies for systems in all dimensions, the significant applications are for d = 2 where it is shown: (a) If the mean–field theory has a discontinuous phase transition featuring the breaking of a discrete symmetry then this sort of transition will occur in the actual system. Prominent examples include the two–dimensional q = 3 state Potts model. (b) If the mean–field theory has a discontinuous transition accompanied by the breaking of a continuous symmetry, the thermodynamic discontinuity is preserved even if the symmetry breaking is forbidden in the actual system. E.g. the two–dimensional O(3) nematic liquid crystal. Further it is demonstrated that mean–field behavior in the vicinity of the magnetic transition for layered Ising and XY systems also occurs in actual layered systems (with spread–out interactions) even if genuine magnetic ordering is precluded.

We study the McKean–Vlasov equation on the finite tori of length scale L in d-dimensions. We derive the necessary and sufficient conditions for the existence of a phase transition, which are based on the criteria first uncovered in Gates and Penrose (Commun. Math. Phys. 17:194–209, 1970) and Kirkwood and Monroe (J. Chem. Phys. 9:514–526, 1941). Therein and in subsequent works, one finds indications pointing to critical transitions at a particular model dependent value, θ ♯ of the interaction parameter. We show that the uniform density (which may be interpreted as the liquid phase) is dynamically stable for θ<θ ♯ and prove, abstractly, that a critical transition must occur at θ=θ ♯ . However for this system we show that under generic conditions—L large, d≥2 and isotropic interactions—the phase transition is in fact discontinuous and occurs at some $\theta_{\text{T}}<\theta^{\sharp }$ . Finally, for H-stable, bounded interactions with discontinuous transitions we show that, with suitable scaling, the $\theta_{\text{T}}(L)$ tend to a definitive non-trivial limit as L→∞.

We establish convergence to SLE6 of the Exploration Processes for the correlated bond-triangular type models studied by two of us in an earlier work (Chayes and Lei in Rev. Math. Phys. 19:511–565, 2007). This (rigorously) establishes that these models are in the same universality class as the standard site percolation model on the triangular lattice. In the context of these models, the result is proven for all domains with boundary Minkowski dimension less than two. Moreover, the proof of convergence applies in the context of general critical 2D percolation models and for general domains, under the stipulation that Cardy’s Formula can be established for domains in this generality.

We show how to extract Cardy’s Formula for a general class of domains given the requisite interior analyticity statement. This is accomplished by a careful study of the interplay between discretization schemes and extraction of limiting boundary values. Of particular importance to the companion work (Binder et al. in J. Stat. Phys., 2010) we establish these results for slit domains and for the critical percolation models introduced in Chayes and Lei (Rev. Math. Phys. 19:511–565, 2007).