Characterization and scaling limit of three-dimensional loop-erased random walk and random-walk loop-catchers

Abstract

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.

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Cite

BibTeX RIS

BibTeX

@misc{cai2026characterization,
  title = {{Characterization and scaling limit of three-dimensional loop-erased random walk and random-walk loop-catchers}},
  author = {Gefei Cai and Xinyi Li and Daisuke Shiraishi},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.12047},
  eprint = {2610.12047},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.12047},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Characterization and scaling limit of three-dimensional loop-erased random walk and random-walk loop-catchers
AU  - Gefei Cai
AU  - Xinyi Li
AU  - Daisuke Shiraishi
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.12047
UR  - https://arxiv.org/abs/2610.12047
AB  - 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.
ER  -