| Budget Amount *help |
¥3,770,000 (Direct Cost: ¥2,900,000、Indirect Cost: ¥870,000)
Fiscal Year 2017: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2016: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2015: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
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| 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.
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