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PreprintCC BY 4.0arXiv

The conformally invariant metric on CLE4 II: existence of geodesics

  • Emmanuel Kammerer
  • Konstantinos Kavvadias
  • Jason Miller
  • Yi Tian
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Abstract

We continue our study of the conformal loop ensemble (CLE) with parameter κ=4, the critical threshold at or below which the loops are simple and disjoint, touching neither each other nor the domain boundary. This paper is the second in a series of three establishing that the loops of a CLE4 uniquely determine a conformally invariant, local, and geodesic metric such that the metric ball growth from the domain boundary coincides with the uniform exploration of Werner and Wu. In this second paper, we prove the existence of geodesics, showing that any geodesic between two loops is supported on the CLE4 loops (off a set of Hausdorff dimension zero) and does not intersect the domain boundary. Along the way, we establish sharp quantitative estimates for the CLE4 metric geometry, including exponential tail bounds for rectangle distances and multi-scale four-arm SLE4 non-intersection bounds for metric balls.

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Cite

BibTeXRIS
BibTeX
@misc{kammerer2026conformally,
  title = {{The conformally invariant metric on CLE$_4$ II: existence of geodesics}},
  author = {Emmanuel Kammerer and Konstantinos Kavvadias and Jason Miller and Yi Tian},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.04139},
  eprint = {2609.04139},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.04139},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The conformally invariant metric on CLE_4 II: existence of geodesics
AU  - Emmanuel Kammerer
AU  - Konstantinos Kavvadias
AU  - Jason Miller
AU  - Yi Tian
PY  - 2026
DA  - 2026/09/03
PB  - arXiv
DO  - 10.48550/arxiv.2609.04139
UR  - https://arxiv.org/abs/2609.04139
AB  - We continue our study of the conformal loop ensemble (CLE) with parameter κ=4, the critical threshold at or below which the loops are simple and disjoint, touching neither each other nor the domain boundary. This paper is the second in a series of three establishing that the loops of a CLE_4 uniquely determine a conformally invariant, local, and geodesic metric such that the metric ball growth from the domain boundary coincides with the uniform exploration of Werner and Wu. In this second paper, we prove the existence of geodesics, showing that any geodesic between two loops is supported on the CLE_4 loops (off a set of Hausdorff dimension zero) and does not intersect the domain boundary. Along the way, we establish sharp quantitative estimates for the CLE_4 metric geometry, including exponential tail bounds for rectangle distances and multi-scale four-arm SLE_4 non-intersection bounds for metric balls.
ER  -