| Project/Area Number |
19K03540
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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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| Review Section |
Basic Section 12010:Basic analysis-related
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| Research Institution | Kyoto University |
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Principal Investigator |
Croydon David 京都大学, 数理解析研究所, 准教授 (50824182)
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| Project Period (FY) |
2019-04-01 – 2024-03-31
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| Project Status |
Completed (Fiscal Year 2023)
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| Budget Amount *help |
¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000)
Fiscal Year 2022: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2021: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2020: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2019: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
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| Keywords | random walks / random graphs / subdiffusive behaviour / uniform spanning trees / Mott random walk / random conductance model / heat kernel estimates / scaling limits / homogenization / percolation / Mott hopping / Extremal process / Random walk / Localization / Random environment / random walk / subdiffusion / trapping / hear kernel estimates / uniform spanning tree / critical dimension / scaling limit / fractals |
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| Outline of Research at the Start |
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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| Outline of Final Research Achievements |
A range of random walks in random environments were studied, many motivated by problems in mathematical physics. One of the important examples for which significant new results were proved was the uniform spanning tree in two and three dimensions, which is a fundamental example of a random tree constrained by Euclidean space and arises as a limit of a certain model in statistical physics. Another model for which new results were obtained was Mott variable range hopping, which is a model for electron transport in inhomogeneous media. In both cases, scaling limits were obtained that show the long-time behaviour of the processes in question. Other models considered included the random conductance model, the Bouchaud trap model and percolation on certain random planar maps. In all cases, a theme of the analysis was the use of resistance forms, which were originally developed in the context of analysis on fractals, but are now being seen to be useful for understanding random environments.
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| Academic Significance and Societal Importance of the Research Achievements |
The motivation for studying random walks in random environments is to provide into the transport properties of disordered media. The results of this project focussed on regimes where anomalous behaviour can be observed, and thus it helps explain what features lead to a break from typical behaviour.
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