2021 Fiscal Year Research-status 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 – 2023-03-31
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| Keywords | random walk / percolation / subdiffusion / trapping / hear kernel estimates |
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| Outline of Annual Research Achievements |
Within this project, the models the PI has focussed on this year include:
1. Biased random walk on supercritical percolation: Together with Adam Bowditch (University College Dublin), the PI demonstrated anomalous fluctuations for the process in the ballistic regime. This is a challenging problem as it involves understanding the "second order" behaviour of the model, which had not previously been studied.
2. Simple random walk on a long-range percolation cluster: Together with Van Hao Can (Vietnam Academy of Science and Technology) and Takashi Kumagai (Waseda University), the PI determined the spectral dimension throughout the range of parameters where the model is defined, apart from at one critical point. This clarifies the impact of long-range connections on the behaviour of random walk.
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
As with last year, due to travel restrictions during the coronavirus pandemic, the PI's plans for research travel this year were seriously curtailed, reducing
opportunities for research dissemination and discussion with potential new collaborators. Nonetheless, the PI continued to work productively with collaborators via zoom, which led to the completion of the projects described above. He also has various ongoing projects related to the original proposal and future travel plans, as will be listed in the "planning for future work" section of this report.
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| Strategy for Future Research Activity |
The PI has a number of ongoing projects, which include:
- Mott variable-range hopping: The PI will work with Ryoki Fukushima (Tsukuba) and Stefan Junk (Tohoku) on an 'extremal' version of this model; and with Daniel Kious and Carlo Scali (both Bath) on an 'aging' property of it.
- Scaling limit of random walk on the incipient infinite cluster of
oriented percolation in high dimensions: Joint with Manuel Cabezas (PUC, Chile), Alex Fribergh (Montreal).
- Annealed heat kernel of random walk on the range of random walk: Joint with Satomi Watanabe and Daisuke Shiraishi (both Kyoto), the project will help clarify the distinction between quenched and annealed heat kernel behaviour.
In the second half of the year, he will travel to the UK to the University of Oxford, where he will work with Ben Hambly on the spectral asymptotics of random fractals. From there, he will also travel to meetings in Bath, Warwick and Oberwolfach.
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| Causes of Carryover |
As noted above, as with last year, travel restrictions this year meant many plans were cancelled/postponed. The PI plans to resume such activities this year now circumstances seem to be allowing it.
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Research Products
(4 results)
