02-325 Marek Biskup and Lincoln Chayes
Rigorous analysis of discontinuous phase~transitions via mean-field bounds (604K, PDF Document) Jul 25, 02
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Abstract. We consider a variety of nearest-neighbor spin models defined on the $d$-dimensional hypercubic lattice~$\Z^d$. 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~$\Z^d$ undergoes a first-order phase transition as the temperature varies. Similar results are established for all~$q$-state Potts models with $q\ge3$, the~$r$-component cubic models with $r\ge4$ and the~$O(N)$-nematic liquid-crystal models with~$N\ge3$.

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