FERROMAGNETIC ISING MEASURES ON LARGE LOCALLY TREE-LIKE GRAPHS

• 成果类型: Article
• 署名作者: Basak, Anirban; Dembo, Amir
• 署名单位: Duke University; Stanford University; Stanford University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/15-AOP1075
• 发表日期: 2017
• 页码: 780-823
• 关键词: phase-transitions; MODEL; percolation; networks; field

摘要

We consider the ferromagnetic Ising model on a sequence of graphs G(n) converging locally weakly to a rooted random tree. Generalizing [Probab. Theory Related Fields 152 (2012) 31-51], under an appropriate continuity property, we show that the Ising measures on these graphs converge locally weakly to a measure, which is obtained by first picking a random tree, and then the symmetric mixture of Ising measures with + and -boundary conditions on that tree. Under the extra assumptions that G(n) are edge-expanders, we show that the local weak limit of the Ising measures conditioned on positive magnetization is the Ising measure with + boundary condition on the limiting tree. The continuity property holds except possibly for countable many choices of beta, which for limiting trees of minimum degree at least three, are all within certain explicitly specified compact interval. We further show the edge-expander property for (most of) the configuration model graphs corresponding to limiting (multi-type) Galton-Watson trees.