Fast initial conditions for Glauber dynamics

• 成果类型: Article
• 署名作者: Lubetzky, Eyal; Sly, Allan
• 署名单位: New York University; Princeton University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-020-01015-3
• 发表日期: 2021
• 页码: 647-667
• 关键词: ising-model; cutoff

摘要

In the study of Markov chain mixing times, analysis has centered on the performance from a worst-case starting state. Here, in the context of Glauber dynamics for the one-dimensional Ising model, we show how new ideas from information percolation can be used to establish mixing times from other starting states. At high temperatures we show that the alternating initial condition is asymptotically the fastest one, and, surprisingly, its mixing time is faster than at infinite temperature, accelerating as the inverse-temperature beta ranges from 0 to beta(0) = 1/2arctanh (1/3). Moreover, the dominant test function depends on the temperature: at beta > beta(0) it is autocorrelation, whereas at beta>beta(0) it is the Hamiltonian.