Glauber dynamics for the mean-field Ising model: cut-off, critical power law, and metastability

• 成果类型: Article
• 署名作者: Levin, David A.; Luczak, Malwina J.; Peres, Yuval
• 署名单位: University of Oregon; University of London; London School Economics & Political Science; Microsoft; University of Washington; University of Washington Seattle; University of California System; University of California Berkeley
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-008-0189-z
• 发表日期: 2010
• 页码: 223-265
• 关键词: markov-chains; limit

摘要

We study the Glauber dynamics for the Ising model on the complete graph, also known as the Curie-Weiss Model. For beta < 1, we prove that the dynamics exhibits a cut-off: the distance to stationarity drops from near 1 to near 0 in a window of order n centered at [2(1 - beta)](-1)n log n. For beta = 1, we prove that the mixing time is of order n(3/2). For beta > 1, we study metastability. In particular, we show that the Glauber dynamics restricted to states of non-negative magnetization has mixing time O(n log n).