2023 Fiscal Year Annual Research Report
Stochastic processes associated with resistance forms
| Project/Area Number |
19K03540
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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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| Keywords | random walks / random graphs / heat kernel estimates / homogenization / percolation |
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| Outline of Annual Research Achievements |
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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