| Project/Area Number |
24540106
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Hokkaido University |
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Principal Investigator |
SAKAI Akira 北海道大学, 理学(系)研究科(研究院), 准教授 (50506996)
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| Project Period (FY) |
2012-04-01 – 2015-03-31
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| Project Status |
Completed (Fiscal Year 2014)
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| Budget Amount *help |
¥5,070,000 (Direct Cost: ¥3,900,000、Indirect Cost: ¥1,170,000)
Fiscal Year 2014: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2013: ¥3,510,000 (Direct Cost: ¥2,700,000、Indirect Cost: ¥810,000)
Fiscal Year 2012: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
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| 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 / イジング模型（オランダ，台湾） / レース展開（台湾） / イジング1-arm指数（オランダ） / 臨界現象 / 1-arm指数（オランダ） / イジング模型（オランダ） |
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| 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.
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