发表机构
Beijing International Center for Mathematical Research, Peking University; Graduate School of Informatics, Kyoto University（北京大学北京国际数学研究中心; 京都大学情报学府）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究刻画三维环消随机游走（LERW）与随机游走环捕获器（RWLC）的标度极限，解决Sapozhnikov-Shiraishi猜想三维情形，证明LERW标度极限存在性，构造三维布朗环捕获器并得到其反演不变性。
AI 中文摘要
随机游走环捕获器（RWLC）是随机游走轨迹的一类单参数随机子集族，由[蔡2026，arXiv:2607.18070]新近提出，它在环消随机游走（LERW）与完整游走轨迹之间插值，满足恢复性质。我们利用纠缠多路径LERW的格林函数检验，证明该恢复性质的连续版本可唯一刻画三维LERW与RWLC的标度极限，相关内容概述于[蔡2026，arXiv:2607.18070]。对于LERW，这解决了Sapozhnikov-Shiraishi猜想的三维情形[2018，Probab. Theory Related Fields 172，615-662]，并为其标度极限的存在性提供了新的公理化证明，该存在性最早由[Kozma 2007，Acta Math. 199，29-152]证明；对于RWLC，这构造了三维布朗环捕获器。该刻画还给出了三维LERW标度极限的反演不变性。
英文摘要
Random-walk loop-catchers (RWLC) are a one-parameter family of random subsets of the trace of a random walk, recently introduced in [Cai 2026, arXiv:2607.18070], which interpolates between loop-erased random walk (LERW) and the full walk trace, satisfying a recovery property. We prove that the continuous version of this recovery property uniquely characterizes the scaling limit of LERW and RWLC in three dimensions, using a Green-function test via entangled multipath LERW, as outlined in [Cai 2026, arXiv:2607.18070]. For LERW, this resolves the 3D case of the conjecture of Sapozhnikov--Shiraishi [Sapozhnikov--Shiraishi 2018, Probab. Theory Related Fields 172, 615--662] and gives a new and axiomatic proof of the existence of the scaling limit, which was first proved in [Kozma 2007, Acta Math. 199, 29--152]. For RWLC, this gives the construction of 3D Brownian loop-catchers. The characterization also yields inversion invariance of the 3D LERW scaling limit.
Comments28 pages
