The self-dual point of the two-dimensional random-cluster model is critical for q ≥ 1

• 成果类型: Article
• 署名作者: Beffara, Vincent; Duminil-Copin, Hugo
• 署名单位: Ecole Normale Superieure de Lyon (ENS de LYON); University of Geneva
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-011-0353-8
• 发表日期: 2012
• 页码: 511-542
• 关键词: critical probability; potts-model; percolation; statistics

摘要

We prove a long-standing conjecture on random-cluster models, namely that the critical point for such models with parameter q a parts per thousand yen 1 on the square lattice is equal to the self-dual point . This gives a proof that the critical temperature of the q-state Potts model is equal to for all q a parts per thousand yen 2. We further prove that the transition is sharp, meaning that there is exponential decay of correlations in the sub-critical phase. The techniques of this paper are rigorous and valid for all q a parts per thousand yen 1, in contrast to earlier methods valid only for certain given q. The proof extends to the triangular and the hexagonal lattices as well.