On yet-open questions about Ising ferromagnets

2017 Fiscal Year Final Research Report

• Project/Area Number: 15K13440
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Basic analysis
• Research Institution: Hokkaido University
• Principal Investigator: Sakai Akira 北海道大学, 理学研究院, 准教授 (50506996)
• Project Period (FY): 2015-04-01 – 2018-03-31
• Keywords: イジング模型 / 臨界現象 / 1-arm指数 / 低温相 / 確率幾何的表現

Outline of Final Research Achievements

Consider the 1-spin expectation at the center of the d-dimensional ball of radius r under the plus-boundary condition. It has been known that it exhibits a phase transition and decays to zero as r diverges at the critical temperature. In particular, it is known to decay slower than r to the power 1-d/2 above the upper-critical dimension 4, due to a hyperscaling inequality.
By using the random-current representation, a stochastic-geometric representation for the Ising model, Handa, Heydenreich and I have proven a sharper second-moment estimate and concluded that the critical 1-spin expectation decays no faster than 1/r above 4 dimensions, meaning the 1-arm exponent is not bigger than the long-expected mean-field value 1 for d>4. The results are summarized in a paper, which was accepted for publication in a festschrift for Charles Newman's 70th birthday.

Free Research Field

確率論，統計力学，数理物理