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2019 Fiscal Year Research-status Report

Stochastic processes associated with resistance forms

Research Project

Project/Area Number 19K03540
Research InstitutionKyoto University

Principal Investigator

Croydon David  京都大学, 数理解析研究所, 准教授 (50824182)

Project Period (FY) 2019-04-01 – 2023-03-31
Keywordsuniform spanning tree / scaling limit / random walk / heat kernel estimates
Outline of Annual Research Achievements

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.

Current Status of Research Progress
Current Status of Research Progress

2: Research has progressed on the whole more than it was originally planned.

Reason

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.

Strategy for Future Research Activity

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.

  • Research Products

    (9 results)

All 2020 2019

All Journal Article (1 results) (of which Int'l Joint Research: 1 results,  Peer Reviewed: 1 results) Presentation (8 results) (of which Int'l Joint Research: 1 results,  Invited: 8 results)

  • [Journal Article] The random conductance model with heavy tails on nested fractal graphs2020

    • Author(s)
      David Croydon
    • Journal Title

      Fractal Geometry and Stochastics

      Volume: VI Pages: -

    • Peer Reviewed / Int'l Joint Research
  • [Presentation] Scaling limits of the two- and three-dimensional uniform spanning trees and the associated random walks2019

    • Author(s)
      David Croydon
    • Organizer
      New York University, Courant Institute, Probability and Mathematical Physics Seminar
    • Invited
  • [Presentation] Random walks on the two- and three-dimensional uniform spanning trees2019

    • Author(s)
      David Croydon
    • Organizer
      Kansai University, International workshop on stochastic analysis and applications
    • Invited
  • [Presentation] Scaling limits of random walks on random graphs in critical regimes2019

    • Author(s)
      David Croydon
    • Organizer
      Kanazawa University, Mathematical Society of Japan autumn meeting
    • Invited
  • [Presentation] Random walks on the two- and three-dimensional uniform spanning trees2019

    • Author(s)
      David Croydon
    • Organizer
      Fukuoka University, Japanese-German open conference on stochastic analysis
    • Invited
  • [Presentation] Random walks on fractals and critical random graphs2019

    • Author(s)
      David Croydon
    • Organizer
      PUC/Universidad de Chile, Probability seminar
    • Invited
  • [Presentation] Quenched and averaged tails of the heat kernel of the two-dimensional uniform spanning tree2019

    • Author(s)
      David Croydon
    • Organizer
      Northwestern University, 41st stochastic processes and their applications conference
    • Int'l Joint Research / Invited
  • [Presentation] Scaling limits of random walks on random graphs in critical regimes2019

    • Author(s)
      David Croydon
    • Organizer
      Kyoto University, Joint colloquium of the Mathematics Department/Research Institute for Mathematical Sciences
    • Invited
  • [Presentation] Quenched and averaged tails of the heat kernel of the two-dimensional uniform spanning tree2019

    • Author(s)
      David Croydon
    • Organizer
      Kobe University, Workshop on probabilistic potential theory and related fields
    • Invited

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Published: 2021-01-27  

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