Rigorous Analysis of Discontinuous Phase Transitions via Mean-Field Bounds

2002/07/31 by Marek Biskup, Lincoln Chayes · 5 citations
Mathematics · Physics and Astronomy · #Random Matrices and Applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math.PR #msc:82B05 #msc:82B20 #msc:82B26

paper · pdf · doi:10.1007/s00220-003-0828-2

published as Commun. Math. Phys. 238 (2003), no. 1-2, 53-93 · 41 pages

openalex publication_date 2003/07/01 · arxiv created 2003/09/30 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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

Citations

• On the critical behavior of the magnetization in high-dimensional Ising models
• The Physics of Liquid Crystals
• Some generalized order-disorder transformations

Cited by

• Liquid-vapor transition in a model of a continuum particle system with finite-range modified Kac pair potential
• Complete analysis of phase transitions and ensemble equivalence for the Curie–Weiss–Potts model
• Limit theorems and coexistence probabilities for the Curie–Weiss Potts model with an external field
• Ferromagnetism, antiferromagnetism, and the curious nematic phase ofS=1quantum spin systems
• Methods of Contemporary Mathematical Statistical Physics

Related

• Lectures on the Ising and Potts models on the hypercubic lattice
• Discontinuity of the phase transition for the planar random-cluster and Potts models with q>4
• Quantitative disorder effects in low-dimensional spin systems
• Swendsen-Wang Algorithm on the Mean-Field Potts Model
• Unexpected results of the phase transitions of four-state Potts model on the square and the honeycomb lattices