{
  "abstract": "Let $W_{N,N+λ}$ have the Laguerre unitary distribution with size $N$ and real shape $λ\\ge0$. For $s>0$, we consider the normalized moment $$C_{s,λ}(N)=N^{-s-1}\\mathbb{E}[\\operatorname{Tr}W_{N,N+λ}^{s}]$$ and its dimension decrement $Γ_{N,s,λ}=C_{s,λ}(N)-C_{s,λ}(N+1)$. Iterating the Laguerre moment recurrence separates this decrement into a square source and a nonnegative shape source. The square source gives the complete finite-dimensional sign diagram: $C_{s,0}(N)$ decreases for $0<s<1$ and $s>2$, increases for $1<s<2$, and is constant for $s\\in\\{1,2\\}$. For every $s>0$ and every $N$, the decrement is strictly increasing in $λ$. The same decomposition determines the critical shrinking-shape scales: $N^{-2s}$ for $0<s<1/2$, $(\\log N)/N$ for $s=1/2$, and $N^{-1}$ for $s>1/2$. In the convex range $1<s<2$, where the two sources have opposite signs, the transition occurs when $Nλ_N$ is of order one, with critical constant $$τ_s^*=\\frac{s(s-1)(2-s)}{6(2s-1)}.$$ In the convex range, both crossings are unique for every finite $N$, and we determine their locations to second order. At $s=1/2$ we also obtain a bounded-shape two-term expansion, which supplies the unitary estimates used in the companion orthogonal paper.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.12147",
  "author_total": 1,
  "authors": [
    {
      "name": "Ondrej Hutník"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/77gc2mq576ygp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.12147",
  "fulltext_url": null,
  "id": "77gc2mq576ygp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.12147",
  "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.PR"
    ],
    [
      "openalex",
      "2613|Random Matrices and Applications"
    ]
  ],
  "openalex": "W7202360777",
  "pdf_url": "https://arxiv.org/pdf/2608.12147",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/77gc2mq576ygp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:11Z",
      "native_id": "2608.12147",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.12147"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Dimension Monotonicity in Laguerre Ensembles I: Fractional Moments and Shape Transitions in the Unitary Case",
  "title_text": "Dimension Monotonicity in Laguerre Ensembles I: Fractional Moments and Shape Transitions in the Unitary Case",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:11Z",
  "url": "https://research.draconitic.com/paper/77gc2mq576ygp",
  "venue": "arXiv",
  "versions": []
}