{
  "abstract": "A permuton is a probability measure on the unit square with uniform marginals, the limit shape of a sequence of permutations. The sticky product of permutons is a continuous analogue of the Demazure product in the $0$-Hecke monoid; on copulas, the distribution functions of permutons, it is given by the tropical $(\\min,+)$ product $\\odot$. We show that every absolutely continuous rotation-invariant probability measure $ρ$ on the plane, which we call the profile, generates a one-parameter family of copulas $C^ρ=\\{C^ρ_β\\}_{β\\in[0,π]}$. A copula in this family is defined by a pair of projections of a $ρ$-distributed random vector onto directions at angle $π-β$. Along this family, the sticky product adds angles: $$C^ρ_{β_1}\\odot C^ρ_{β_2}=C^ρ_{\\min(β_1+β_2,π)}.$$ Hence every non-identity element of such a family is a sticky product of two others and has roots of every natural degree. The normalized inversion number of the copula $C^ρ_β$ equals $β/π$. For the Archimedean measure $\\mathfrak{Arch}$ the family $C^{\\mathfrak{Arch}}$ consists of the limit shapes of the permutations of the uniformly random sorting network, found by Angel et al. and Dauvergne. For the Gaussian measure $\\mathcal{N}$ the family $C^{\\mathcal{N}}$ consists of Gaussian copulas and contains $\\mathit{Unif}$, the uniform measure on the square (the limit shape of the uniformly random permutation); its square root within this family is the Gaussian copula with correlation $-1/\\sqrt2$. By the Herschel-Maxwell theorem the Gaussian measure is the only profile whose family contains $\\mathit{Unif}$; nevertheless, $\\mathit{Unif}$ has another, singular, sticky square root. A discrete analogue of the Gaussian family is the geometric sorting network generated by rotating independent Gaussian points: after rotation by $π/2$ its permutation is exactly uniform.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.26700",
  "author_total": 2,
  "authors": [
    {
      "name": "Sergei Presniakov"
    },
    {
      "name": "Alexander Tiskin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0g50j7091ayjz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.26700",
  "fulltext_url": null,
  "id": "0g50j7091ayjz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.26700",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7223066832",
  "pdf_url": "https://arxiv.org/pdf/2610.26700",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0g50j7091ayjz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:31:00Z",
      "native_id": "2610.26700",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.26700"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Geodesics in the continuous Hecke monoid",
  "title_text": "Geodesics in the continuous Hecke monoid",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:47Z",
  "url": "https://research.draconitic.com/paper/0g50j7091ayjz",
  "venue": "arXiv",
  "versions": []
}