On trace of Brownian motion and related models from statistical physics

2022 Fiscal Year Final Research Report

• Project/Area Number: 18K13425
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 12010:Basic analysis-related
• Research Institution: Kyoto University
• Principal Investigator: Shiraishi Daisuke 京都大学, 情報学研究科, 准教授 (00647323)
• Project Period (FY): 2018-04-01 – 2023-03-31
• Keywords: ランダムウォーク

Outline of Final Research Achievements

In this research we consider loop-erased random walk (LERW) and its scaling limit in three dimensions, and prove that 3D LERW parametrized by renormalized length converges to its scaling limit parametrized by some suitable measure with respect to the uniform convergence topology in the lattice size scaling limit. Our result greatly improves the work of Gady Kozma which establishes the weak convergence of the rescaled trace of 3D LERW towards a random compact set with respect to the Hausdorff distance. To prove this, we also need to give an asymptotic estimate on the one-point function for LERW and the non-intersection probability of LERW and simple random walk in three dimensions for dyadic scales. These estimates will be crucial to the characterization of the convergence of LERW to its scaling limit in natural parametrization. As a step in the proof, we also obtain a coupling of two pairs of LERW and SRW with different starting points conditioned to avoid each other.

Free Research Field

確率論

Academic Significance and Societal Importance of the Research Achievements

ループ除去ランダムウォークは他の多くの確率モデルと関わりのある重要な非マルコフ過程であるが、特に３次元の場合は最も解析が困難であり、研究があまり進んでいない状況であった。2次元の場合のように複素解析を用いた共形場理論の手法を用いたり、高次元の場合のような平均場の理論を適用することが出来ないことが本質的な困難を生む。本研究では、離散調和解析、関数解析、確率論、幾何学といった分野横断的な手法を組み合わせることにより、一つ一つの困難を地道に対処していった。そこで培われた手法は、他の次元の場合に対しても適用可能である。