2020 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 / uniform spanning tree / critical dimension / Mott random walk / heat kernel estimates / scaling limits |
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| Outline of Annual Research Achievements |
Within this project, the models the PI has focussed on this year include:
1. Two-dimensional uniform spanning tree: As indicated in the report for the previous year, together with Martin Barlow (UBC) and Takashi Kumagai (Kyoto), the PI completed an article on the heat kernel of the the RW on this object. The estimates reveal an interesting discrepancy between the quenched and averaged exponents, not seen in previous studies.
2. Mott variable-range hopping: Together with Ryoki Fukushima and Stefan Junk (both Tsukuba), the PI completed an article in which a scaling limit for a one-dimensional version of this model was obtained. This clarifies the behaviour of the process in the sub-diffusive regime, which had previously not been explored.
3. Four-dimensional random walk on the range of random walk. In a joint work with Daisuke Shiraishi (Kyoto), the PI derives scaling limits for the random walk whose state space is the range of a simple random walk on the four-dimensional integer lattice. Higher-dimensional versions of the problem had previously been studied by the PI. The PI believes that this work is a first application of 'resistance form' theory to a model at its critical dimension, where logarithmic terms appear in the scaling factors.
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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
Due to travel restrictions during the coronavirus pandemic, the PI's plans for research travel this year were seriously curtailed. In particular, visits to Banff, Cornell and Kyushu were all cancelled, and other proposed trips were also not undertaken. Naturally, this reduced opportunities for research dissemination and discussion with potential new collaborators.
At the same time, as is reported above, the PI made good progress with established collaborators. Both uniform spanning trees and the Mott random walk are explicitly listed within the original proposal. Moreover, the four-dimensional random walk on the range of random walk is closely connected to the original aims.
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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 and Stefan Junk (both Tsukuba) on an 'extremal' version of this model; Random walk on long-range percolation: Together with Takashi Kumagai (Kyoto) and Van Hao Can (Vietnam Academy of Sciences), the PI is attempting to derive heat kernel estimates for models of this type.
The following projects were listed in the plan of the previous year, and are still ongoing:- Together with Omer Angel (UBC), Sarai Hernandez-Torres (UBC) and Daisuke Shiraishi (Kyoto), the PI is studying topological properties of the three-dimensional UST; Together with Adam Bowditch (UCD), the PI has a project concerning the trapping of biased random walk on a supercritical percolation cluster in the ballistic, sub-Gaussian regime; Together with Manuel Cabezas (PUC, Chile), the PI has a project on the scaling limit of random walk on the incipient infinite cluster of oriented percolation in high dimensions.
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| Causes of Carryover |
As noted above, travel restrictions this year meant many plans were cancelled/postponed. The PI plans to resume such activities this year when circumstances allow.
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