Rigorous analysis of discontinuous phase transitions via mean-field bounds

Details Rigorous analysis of discontinuous phase transitions via mean-field bounds Rigorous analysis of discontinuous phase transitions via mean-field bounds

2002-07-25

arxiv.org/abs/math/0207242v2

Commun. Math. Phys. 238 (2003), no. 1-2, 53-93

Mathematics

Probability

41 pages

Scientific paper

10.1007/s00220-003-0828-2

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>=3, the r-component cubic models with r>=4 and the O(N)-nematic liquid-crystal models with N>=3.

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