2000 Fiscal Year Final Research Report Summary
Mathematical Study of Critical Phenomena for Statistical Models in Probability
| Project/Area Number |
11640104
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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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 | Nagoya University (2000)
Tokyo Institute of Technology (1999) |
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Principal Investigator |
HARA Takashi Nagoya University, Graduate School of Mathematics, Associate Professor, 大学院・多元数理科学研究科, 助教授 (20228620)
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| Co-Investigator(Kenkyū-buntansha) |
UCHIYAMA Kohei Tokyo Institute of Technology, Graduate School of Science and Engineering, Professor, 大学院・理工学研究科, 教授 (00117566)
SHIGA Tokuzo Tokyo Institute of Technology, Graduate School of Science and Engineering, Professor, 大学院・理工学研究科, 教授 (60025418)
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| Project Period (FY) |
1999 – 2000
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| Keywords | percolation / critical phenomena / critical point / continuum limit / scaling limit / infinite cluster / integrated super-Brownian excursion |
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| Research Abstract |
The main purpose of the research was to investigate critical behavior of certain stochastic geometric models that appear in probability theory and statistical mechanics. In particular, we investigated continuum (scaling) limits of incipient infinite clusters of the critical percolation model in high dimensions.
We found that a reasonable continuum limit could be obtained for a cluster of size N, if we scale the space proportional to the fourth root of N.Moreover, we calculated the first and second moments of the limiting distributions, and identified them as the first and second moments of the integrated super-Brownian excursion (ISE). This strongly suggests that the continuum limit is in fact ISE.
The method of proof uses the lace expansion, which has been used successfully in other contexts. To investigate the particular problem of this research, we calculated generating functions of cluster distributions, and applied Tauberian analysis.
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