{
  "abstract": "This paper establishes a mesoscopic central limit theorem for linear statistics of $β$-ensembles or log-gas, as the dimension $N\\to\\infty$, in the temperature regime $1/N\\llβ(N)\\le 1$. For simplicity, we assume that the potential is one-cut regular and analytic. In this regime, the size of the fluctuations depends on $β(N)$ and the mesoscopic scale. We show that there is a transition at a critical $η\\asymp 1/ Nβ(N)$ between a Random Matrix regime, where the limiting variance is given by the $\\mathsf{H}^{1/2}$-norm and a Poisson regime where the limiting variance is given by the $L^2$-norm. We also describe the critical regime. The proof of the CLT relies on optimal local laws at intermediate temperatures and Stein's method for $β$-ensembles. In particular, in this regime, it is necessary to construct new correction terms to the classical equilibrium measure to obtain a suitable re-centring of linear statistics and describe their fluctuations. We also obtain a free energy expansion.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.25301",
  "author_total": 3,
  "authors": [
    {
      "name": "Charlie Dworaczek Guera"
    },
    {
      "name": "Gaultier Lambert"
    },
    {
      "name": "Luke Peilen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/46bkpdrch6jrv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.25301",
  "fulltext_url": null,
  "id": "46bkpdrch6jrv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.25301",
  "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.MP"
    ],
    [
      "arxiv",
      "math.PR"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.25301",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/46bkpdrch6jrv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:39Z",
      "native_id": "2609.25301",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.25301"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Mesoscopic transition for $β$-ensembles at intermediary temperature",
  "title_text": "Mesoscopic transition for β-ensembles at intermediary temperature",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/46bkpdrch6jrv",
  "venue": "arXiv",
  "versions": []
}