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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