The conformally invariant metric on CLE_4 III: uniqueness

Abstract

This paper is the third and final article in a series of papers constructing the canonical conformally invariant metric on the set of loops of the conformal loop ensemble (CLE) with critical parameter κ=4. The previous two articles construct, as a subsequential limit of the renormalized graph metric on the loops of CLE_κ as κ↓ 4, a conformally invariant, local metric on the loops of a CLE₄ whose metric ball growth from the domain boundary coincides with the uniform exploration of Werner and Wu. In this paper, we establish that this metric is uniquely characterized by its properties, as are its geodesics, and that it is a measurable function of the CLE₄. In particular, we show that the renormalized CLE_κ graph metric converges as κ↓ 4 without passing to a subsequence. A key step in the proof is to show that the metric is determined by the geodesics from each loop to the domain boundary, which are in turn determined by the uniform exploration; this representation will have important applications in future work.

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BibTeX RIS

BibTeX

@misc{kammerer2026conformally,
  title = {{The conformally invariant metric on CLE$_4$ III: uniqueness}},
  author = {Emmanuel Kammerer and Konstantinos Kavvadias and Jason Miller and Yi Tian},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.04140},
  eprint = {2609.04140},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.04140},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - The conformally invariant metric on CLE_4 III: uniqueness
AU  - Emmanuel Kammerer
AU  - Konstantinos Kavvadias
AU  - Jason Miller
AU  - Yi Tian
PY  - 2026
DA  - 2026/09/03
PB  - arXiv
DO  - 10.48550/arxiv.2609.04140
UR  - https://arxiv.org/abs/2609.04140
AB  - This paper is the third and final article in a series of papers constructing the canonical conformally invariant metric on the set of loops of the conformal loop ensemble (CLE) with critical parameter κ=4. The previous two articles construct, as a subsequential limit of the renormalized graph metric on the loops of CLE_κ as κ↓ 4, a conformally invariant, local metric on the loops of a CLE_4 whose metric ball growth from the domain boundary coincides with the uniform exploration of Werner and Wu. In this paper, we establish that this metric is uniquely characterized by its properties, as are its geodesics, and that it is a measurable function of the CLE_4. In particular, we show that the renormalized CLE_κ graph metric converges as κ↓ 4 without passing to a subsequence. A key step in the proof is to show that the metric is determined by the geodesics from each loop to the domain boundary, which are in turn determined by the uniform exploration; this representation will have important applications in future work.
ER  -

The conformally invariant metric on CLE₄ III: uniqueness

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