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PreprintCC BY 4.0arXiv

Monochromatic arm exponents for critical planar percolation

  • Gefei Cai
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Abstract

We determine the monochromatic three-arm exponent for critical planar percolation through the exact spectral threshold of an explicit positive integral operator family with elementary kernels, obtaining ξ3=0.76928375⋯. Unlike the monochromatic two-arm (backbone) event, the three-arm event can be obstructed by two distinct white clusters, so a Markovian exploration must retain the past connections. We introduce a two-point boundary observable that records these connections. Using the coupling between conformal loop ensemble (CLE) and Liouville quantum gravity (LQG), this observable satisfies a renewal equation in terms of quantum boundary lengths, and yields the explicit operator whose spectral information determines ξ3. Our approach also gives an explicit spectral characterization of monochromatic n-arm exponents for all n≥4.

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Cite

BibTeXRIS
BibTeX
@misc{cai2026monochromatic,
  title = {{Monochromatic arm exponents for critical planar percolation}},
  author = {Gefei Cai},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.05101},
  eprint = {2611.05101},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.05101},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Monochromatic arm exponents for critical planar percolation
AU  - Gefei Cai
PY  - 2026
DA  - 2026/11/03
PB  - arXiv
DO  - 10.48550/arxiv.2611.05101
UR  - https://arxiv.org/abs/2611.05101
AB  - We determine the monochromatic three-arm exponent for critical planar percolation through the exact spectral threshold of an explicit positive integral operator family with elementary kernels, obtaining ξ_3=0.76928375⋯. Unlike the monochromatic two-arm (backbone) event, the three-arm event can be obstructed by two distinct white clusters, so a Markovian exploration must retain the past connections. We introduce a two-point boundary observable that records these connections. Using the coupling between conformal loop ensemble (CLE) and Liouville quantum gravity (LQG), this observable satisfies a renewal equation in terms of quantum boundary lengths, and yields the explicit operator whose spectral information determines ξ_3. Our approach also gives an explicit spectral characterization of monochromatic n-arm exponents for all n≥4.
ER  -