| 研究実績の概要 |
This academic year, the main achievements that have been made on this project relate to the uniform spanning tree (UST) in two and three dimensions. (The uniform spanning tree is a natural model in probability theory, with connections to combinatorics, electrical potential theory and statistical mechanics, amongst other areas.) With regards to the three-dimensional case, the PI has submitted two papers jointly with Omer Angel (UBC), Sarai Hernandez-Torres (UBC) and Daisuke Shiraishi (Kyoto). The first of these concerns the scaling limit of the three-dimensional UST, and the associated random walk. The second, shorter article, applies the same techniques to answer a question in the literature concerning the number of spanning clusters of the object in question. In the two dimensional case, together with Martin Barlow (UBC) and Takashi Kumagai (Kyoto), the PI has continued his study of the random walk on the UST. In particular, he has derived detailed heat kernel estimates for the process, which shed light on the interplay between the random geometry and the stochastic process that evolves within this. This study, which will likely be completed soon, includes new estimates for the related model of loop-erased random walk.
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| 現在までの達成度 (区分) |
現在までの達成度 (区分)
2: おおむね順調に進展している
理由
The research on this project has progressed smoothly so far. The results on the uniform spanning tree, as described above, were a key part of the original plan. Moreover, during the research visits and meetings that the PI undertook this year, initial steps have been taken on a number of further projects, as will be detailed in the “Plans for the Research Scheme” below.
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| 今後の研究の推進方策 |
Arising from discussions held in the 2019-20 academic year, the PI has ongoing projects on the following topics, which he plans to continue working on this year and beyond.
- Together with Stefan Junk (a JSPS postdoc based at Kyoto) and Ryoki Fukushi (Tsukuba), he has started work on the Mott variable range hopping model, which attempts to capture the low-temperature behaviour of conductivity in disordered solids. In particular, they will study the scaling limit of a symmetric version of the model in a regime where anomalous limiting processes arise, and also explore in detail the behaviour of the corresponding model with drift.
- Together with Omer Angel (UBC), Sarai Hernandez-Torres (UBC) and Daisuke Shiraishi (Kyoto), he will study topological properties of the three-dimensional UST.
- Together with Adam Bowditch (NUS), he 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), he 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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