Rigorous Analysis of Discontinuous Phase Transitions via Mean-field Bounds

We consider a variety of nearest-neighbor spin models defined on the d-dimensional hypercubic lattice Z. Our essential assumption is that these models satisfy the condition of reflection positivity. We prove that whenever the associated mean-field theory predicts a discontinuous transition, the actual model also undergoes a discontinuous transition (which occurs near the mean-field transition temperature), provided the dimension is sufficiently large or the first-order transition in the mean-field model is sufficiently strong. As an application of our general theory, we show that for d sufficiently large, the 3-state Potts ferromagnet on Z undergoes a first-order phase transition as the temperature varies. Similar results are established for all q-state Potts models with q ≥ 3, the r-component cubic models with r ≥ 4 and the O(N)-nematic liquid-crystal models with N ≥ 3.