The conformally invariant metric on CLE₄ I: subsequential limits of the non-simple CLE graph metric

@misc{kammerer2026conformally,
  title = {{The conformally invariant metric on CLE$_4$ I: subsequential limits of the non-simple CLE graph metric}},
  author = {Emmanuel Kammerer and Konstantinos Kavvadias and Jason Miller and Yi Tian},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.04138},
  eprint = {2609.04138},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.04138},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - The conformally invariant metric on CLE_4 I: subsequential limits of the non-simple CLE graph metric
AU  - Emmanuel Kammerer
AU  - Konstantinos Kavvadias
AU  - Jason Miller
AU  - Yi Tian
PY  - 2026
DA  - 2026/09/03
PB  - arXiv
DO  - 10.48550/arxiv.2609.04138
UR  - https://arxiv.org/abs/2609.04138
AB  - We consider the conformal loop ensemble (CLE) with the parameter κ=4, the critical value at or below which the loops are simple and do not intersect each other or the domain boundary. We show that the loops of a CLE_4 uniquely determine a conformally invariant, local, and geodesic metric so that the metric ball growth from the domain boundary coincides with the uniform exploration of Werner and Wu. This metric was previously constructed in unpublished work of Sheffield, Watson, and Wu. Our approach differs in that we show that the metric arises as the renormalized limit of the graph metric on CLE_κ loops as κ↓ 4. In this first paper in a series of three, we prove that the subsequential limits exist and define a non-trivial conformally invariant metric on CLE_4 which is local and such that the metric ball growth from the boundary is given by the uniform exploration of Werner and Wu. In subsequent work, we will show that the subsequential limit exists as a true limit.
ER  -