Analysis of critical behavior for spin systems using stochastic-geometrical representations

2014 Fiscal Year Final Research Report

• Project/Area Number: 24540106
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: Hokkaido University
• Principal Investigator: SAKAI Akira 北海道大学, 理学(系)研究科(研究院), 准教授 (50506996)
• Project Period (FY): 2012-04-01 – 2015-03-31
• Keywords: スピン系臨界現象 / φ4乗モデル / イジング模型 / レース展開 / イジング1-arm指数

Outline of Final Research Achievements

The (ferromagnatic) Ising model and the φ4 model are known to exhibit phase transition and critical behavior. In 2007, Sakai used a stochastic-geometrical representation, known as the random-current representation, to develop the lace expansion for the Ising model. Extending the use of this stochastic-geometrical representation, we applied the lace expansion to the φ4 model and obtained an asymptotic expression of the critical two-point function in high dimensions. We also established the method of analyzing critical behavior for the models defined by power-law decaying pair potentials, and proved that the critical two-point function in high dimensions is asymptotically Newtonian or Riesz, depending on the value of the power exponent of the pair potentials.

Free Research Field

確率論，統計力学，数理物理