| 研究課題/領域番号 |
19K03540
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| 研究種目 |
基盤研究(C)
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| 配分区分 | 基金 |
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| 応募区分 | 一般 |
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| 審査区分 |
小区分12010:基礎解析学関連
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| 研究機関 | 京都大学 |
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研究代表者 |
Croydon David 京都大学, 数理解析研究所, 准教授 (50824182)
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| 研究期間 (年度) |
2019-04-01 – 2024-03-31
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| 研究課題ステータス |
完了 (2023年度)
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| 配分額 *注記 |
4,290千円 (直接経費: 3,300千円、間接経費: 990千円)
2022年度: 650千円 (直接経費: 500千円、間接経費: 150千円)
2021年度: 1,170千円 (直接経費: 900千円、間接経費: 270千円)
2020年度: 1,300千円 (直接経費: 1,000千円、間接経費: 300千円)
2019年度: 1,170千円 (直接経費: 900千円、間接経費: 270千円)
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| キーワード | random walks / random graphs / heat kernel estimates / homogenization / percolation / Mott hopping / Extremal process / Random walk / Localization / Random environment / random walk / subdiffusion / trapping / hear kernel estimates / uniform spanning tree / critical dimension / Mott random walk / scaling limits / scaling limit / fractals |
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| 研究開始時の研究の概要 |
The main aim of the project is to identify examples of random walks on random graphs to which resistance form techniques can be applied to deduce scaling limits, and derive detailed properties of the limiting processes. Specifically, the PI will consider models such as percolation clusters and uniform spanning trees, biased random walk, and the Mott variable range jump process. Proerties of the random walks and limiting diffusions considered will be heat kernel estimates, cover times and trapping phenomena.
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| 研究実績の概要 |
Amongst the projects I completed this year was a study of the scaling limit of critical percolation clusters on hyperbolic random half-planar triangulations and the associated random walks, which was joint with Eleanor Archer. This model captures the scaling behaviour of high-dimensional critical percolation clusters, and demonstrates the robustness of the resistance form techniques. I also derived annealed transition density estimates for simple random walk on a high-dimensional loop-erased random walk with Daisuke Shiraishi and Satomi Watanabe, which is a test case for more complex models where such estimates are of interest. A key point of note in the result was the difference seen in the quenched and annealed exponents, which supports a conjecture made in the case of the low-dimensional uniform spanning tree. Finally, with Sebastian Andres and Takashi Kumagai, I explored heat kernel fluctuations and quantitative homogenization for the one-dimensional Bouchaud trap model. We expect the basic argument for deriving a quantitiative local limit theorem to be applicable to other instances of random media.
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