FLUCTUATIONS IN MEAN-FIELD ISING MODELS

• 成果类型: Article
• 署名作者: Deb, Nabarun; Mukherjee, Sumit
• 署名单位: Columbia University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/22-AAP1857
• 发表日期: 2023
• 页码: 1961-2003
• 关键词: sparse graph convergence; l-p theory; steins method; approximation; statistics; SEQUENCES

摘要

In this paper, we study the fluctuations of the average magnetization in an Ising model on an approximately dN regular graph GN on N vertices. In particular, if GN satisfies a spectral gap condition, we show that when-ever dN >> ./N, the fluctuations are universal and the same as that of the Curie-Weiss model in the entire ferromagnetic parameter regime. We give a counterexample to demonstrate that the condition dN >> ./N is tight, in the sense that the limiting distribution changes if dN similar to ./N except in the high temperature regime. By refining our argument, we extend universality in the high temperature regime up to dN >> N1/3. Our results include uni-versal fluctuations of the average magnetization in Ising models on regular graphs, Erdos-Renyi graphs (directed and undirected), stochastic block mod-els, and sparse regular graphons. In fact, our results apply to general matrices with nonnegative entries, including Ising models on a Wigner matrix, and the block spin Ising model. As a by-product of our proof technique, we ob-tain Berry-Esseen bounds for these fluctuations, exponential concentration for the average of spins, tight error bounds for the mean-field approximation of the partition function, and tail bounds for various statistics of interest.