| 研究実績の概要 |
Amongst the projects I completed this year was a study of the scaling limit of critical percolation clusters on hyperbolic random half-planar triangulations and the associated random walks, which was joint with Eleanor Archer. This model captures the scaling behaviour of high-dimensional critical percolation clusters, and demonstrates the robustness of the resistance form techniques. I also derived annealed transition density estimates for simple random walk on a high-dimensional loop-erased random walk with Daisuke Shiraishi and Satomi Watanabe, which is a test case for more complex models where such estimates are of interest. A key point of note in the result was the difference seen in the quenched and annealed exponents, which supports a conjecture made in the case of the low-dimensional uniform spanning tree. Finally, with Sebastian Andres and Takashi Kumagai, I explored heat kernel fluctuations and quantitative homogenization for the one-dimensional Bouchaud trap model. We expect the basic argument for deriving a quantitiative local limit theorem to be applicable to other instances of random media.
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