{
  "abstract": "We study global minimizers in $\\mathbb{R}^d$ of the free energy $\\mathcal{E}_κ=\\mathcal{W}+κ\\mathcal{U}$, where $\\mathcal{W}$ is a power-law interaction energy with mild repulsion and $\\mathcal{U}$ is the porous-medium internal energy with exponent $m>1$. In a range of exponents that we call the strict simplex regime, results of Lim--McCann and Davies--Lim--McCann show that $\\mathcal{W}$ is minimized, uniquely up to rigid motions, by equal point masses at the vertices of a regular unit $d$-simplex. Diffusion excludes such atomic minimizers for every $κ>0$. We prove that uniqueness nevertheless persists: for every fixed $m>1$ and all sufficiently small $κ>0$, the free energy has a unique global minimizer up to rigid motions. After a suitable rigid motion, this minimizer is invariant under every symmetry of the simplex, and its support splits into $d+1$ clusters near the vertices, each of mass exactly $1/(d+1)$; in particular, it is not radially symmetric when $d\\geq2$. The clusters have width of order $κ^{1/(d(m-1)+2)}$, which is also the exact order of the quadratic Wasserstein distance from the minimizer to the atomic simplex measure. Rescaled at this scale, the clusters converge in $L^1$ and $L^\\infty$ to explicit Barenblatt-type profiles supported on ellipsoids determined by the Hessian of the simplex potential. For $d\\geq2$ these ellipsoids are spheroids flattened along the directions of the vertices; when the repulsive exponent exceeds two, their aspect ratio equals $(d+1)^{-1/2}$, independently of all other parameters. At the endpoint $(4,2)$ of this regime in dimensions $d\\geq2$, where the zero-diffusion minimizer is not unique, we show instead that vanishing diffusion selects the uniform spherical shell.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.02890",
  "author_total": 2,
  "authors": [
    {
      "name": "Tongseok Lim"
    },
    {
      "name": "Geuntaek Seo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8qwsm4kdv3w76.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.02890",
  "fulltext_url": null,
  "id": "8qwsm4kdv3w76",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.02890",
  "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-ph"
    ],
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.MP"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.02890",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8qwsm4kdv3w76.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:20:59Z",
      "native_id": "2611.02890",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.02890"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Uniqueness and shape of simplex ground states for aggregation--diffusion energies with mild repulsion",
  "title_text": "Uniqueness and shape of simplex ground states for aggregation--diffusion energies with mild repulsion",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/8qwsm4kdv3w76",
  "venue": "arXiv",
  "versions": []
}