{
  "abstract": "We establish a compactness theorem for the resolvent of free Laplacians. More precisely, let $X=(X_1,\\ldots,X_n)$ be a self-adjoint tuple admitting Lipschitz conjugate variables in the sense of Dabrowski (2014). Under the non-commutative curvature--dimension condition $CD(c,\\infty)$ with $c>0$, introduced in our previous work (2026), we prove that the free Laplacian $Δ:=\\partial^*\\bar\\partial$ is proper, that is, has compact resolvent. Our argument is non-perturbative and uses a spectral variant of the free Bochner method based on Dabrowski's almost-commutation formulas. A further stochastic argument gives a normal unital trace-preserving embedding of $M=W^*(X_1,\\ldots,X_n)$ into $L(\\mathbb F_2)$. Together, the compactness and embedding theorems gives several operator-algebraic consequences, including the Haagerup approximation property, the CMAP, and strong solidity. We also obtain proper proximality and bi-exactness. We finally obtain several geometric consequences. We prove a free analogue of Klartag's improved log--Lichnerowicz inequality (2023), together with rigidity in the equality case. More surprisingly, under $CD(0,\\infty)$, we prove a dimension-free Kannan--Lovász--Simonovits (KLS) bound in the free setting. As a consequence, we obtain a free thin-shell estimate for isotropic tuples. We conclude with further questions on spectral comparison with the canonical semicircular model.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.22109",
  "author_total": 1,
  "authors": [
    {
      "name": "Charles-Philippe Diez"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4thab2s13rsc9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.22109",
  "fulltext_url": null,
  "id": "4thab2s13rsc9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.22109",
  "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.OA"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "arxiv",
      "math.SP"
    ],
    [
      "openalex",
      "2610|Advanced Operator Algebra Research"
    ]
  ],
  "openalex": "W7222621274",
  "pdf_url": "https://arxiv.org/pdf/2610.22109",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4thab2s13rsc9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.22109",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.22109"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Convexity, the spectrum of free Laplacians and geometric consequences: a free KLS bound",
  "title_text": "Convexity, the spectrum of free Laplacians and geometric consequences: a free KLS bound",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/4thab2s13rsc9",
  "venue": "arXiv",
  "versions": []
}