2023 Fiscal Year Final Research Report
Stochastic processes associated with resistance forms
| 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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| Keywords | random walks / random graphs / subdiffusive behaviour / uniform spanning trees / Mott random walk / random conductance model / heat kernel estimates / scaling limits |
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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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| Free Research Field |
Probability theory
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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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