{
  "abstract": "We solve an extremal problem for positive randomization kernels that concentrate a random time as tightly as possible around a deterministic target, within the class obtained by exponentially damping the square of a polynomial. The construction, borrowed from the concentrated matrix-exponential literature, automatically guarantees nonnegativity, gives the kernel a Laplace transform with a single repeated real pole, and contains the classical Erlang randomizer as the special case in which the polynomial is a pure power. The half-polynomial need not be real-rooted and so may have nonreal conjugate zeros, yet every optimizer is proved to be real-rooted. After normalizing the kernel to unit mean, the minimum variance at polynomial degree $m$ turns out to equal the smallest relative gap between adjacent zeros of the Laguerre polynomial $L_{m+2}$, with every optimizer obtained by deleting a pair attaining this minimum. This characterization yields the minimal variance, the normalized density, and every optimizer in closed form. The optimizer can be computed from the spectrum of a symmetric tridiagonal Jacobi matrix. In the large-order limit, the minimal variance decays quadratically in the order of the kernel, a marked improvement over the linear decay rate of the Erlang benchmark, and the deleted pair of zeros localizes at normalized location $2$, with limiting absolute separation $2π$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.29143",
  "author_total": 2,
  "authors": [
    {
      "name": "Maria Laura Battagliola"
    },
    {
      "name": "Oscar Peralta"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/883q301dbc3g6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.29143",
  "fulltext_url": null,
  "id": "883q301dbc3g6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.29143",
  "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.CA"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2604|Mathematical functions and polynomials"
    ]
  ],
  "openalex": "W7204922843",
  "pdf_url": "https://arxiv.org/pdf/2608.29143",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/883q301dbc3g6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.29143",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.29143"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Least Variability in a Polynomial-Square Class of Rational Kernels",
  "title_text": "Least Variability in a Polynomial-Square Class of Rational Kernels",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/883q301dbc3g6",
  "venue": "arXiv",
  "versions": []
}