{
  "abstract": "The spectral sample and pivotal set of a percolation crossing have the same one- and two-coordinate inclusion probabilities under the uniform product measure. We prove that both Shannon entropies are comparable to total influence for critical square-lattice bond and triangular-lattice site crossings, with matching estimates at every retained nonroot spatial resolution. The triangular entropy bounds are uniform over a class of inhomogeneous near-critical product measures. For critical triangular-site square crossings, the two laws are asymptotically mutually singular: an elementary triangular face is forbidden in the pivotal set and occurs with positive density in a large spectral sample. Every fixed face density below an explicit threshold has nonempty spectral probability comparable to $R^2α_4(R)^2$, while the nonempty overlap of the two laws has decay exponent $11/12$. Their minimum expected symmetric-difference distance over all couplings is comparable to total influence. Geometric separation persists after independent thinning whenever the retention probability $ρ_R$ satisfies $ρ_R^3R^2α_4(R)\\to\\infty$; the face detector fails when this quantity tends to zero. The proofs combine spatial encoding, arm estimates, local Fourier cancellation, and an exact six-cycle coercivity inequality preserved under exterior conditioning.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.15388",
  "author_total": 1,
  "authors": [
    {
      "name": "Yitzchak Shmalo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9jx8njqk97ecy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.15388",
  "fulltext_url": null,
  "id": "9jx8njqk97ecy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.15388",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2610|Stochastic processes and statistical mechanics"
    ]
  ],
  "openalex": "W7213296107",
  "pdf_url": "https://arxiv.org/pdf/2609.15388",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9jx8njqk97ecy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.15388",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.15388"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Entropy and singularity of spectral and pivotal sets in planar percolation",
  "title_text": "Entropy and singularity of spectral and pivotal sets in planar percolation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/9jx8njqk97ecy",
  "venue": "arXiv",
  "versions": []
}