{
  "abstract": "Randomly connected networks are ubiquitous in computer science, physics or biology. A central question is whether, and how, their collective dynamics depend on the choice of the distribution of the random couplings. Forty years ago, Geman and Hwang formulated a precise version of this \\emph{universality} question for randomly connected neural networks. We come back to these conjectures, review what was proven since, provide a few new extensions and propose new conjectures in this paper. For $n$ neuron networks with couplings with a mean scaling as $1/n$ and light enough tails, two classes of limits are expected. For \\emph{weakly heterogeneous} couplings, with variance $o(1/n)$, the network is predicted to converge to a McKean--Vlasov equation, and a broad literature has established this under general assumptions. For \\emph{strongly heterogeneous} couplings, with variance of order $1/n$, the limit is a non-Markovian equation driven by a self-consistent Gaussian process. After landmark results for Gaussian couplings, universality was only recently established for linear interactions, while the continuous-time nonlinear case relevant to neuroscience remained open. Here, in the weakly heterogeneous regime, we prove quenched convergence on every time interval under a fourth-moment assumption only, and convergence in probability under an assumption slightly stronger than a second moment. In the strongly heterogeneous regime, we prove convergence for nonlinear interactions in continuous time, averaged over sub-Gaussian couplings and on every time interval, by a relative-entropy comparison with Gaussian couplings. We then extend the conjecture to heavy-tailed couplings, and propose a limit driven by a self-consistent $α$-stable process, for which we prove existence for every $α\\in(0,2)$ and uniqueness for $α\\in[1,2)$, and provide extensive numerical evidence in all three cases.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.38893",
  "author_total": 1,
  "authors": [
    {
      "name": "Jonathan Touboul"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/284bd0nxkxa1m.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.38893",
  "fulltext_url": null,
  "id": "284bd0nxkxa1m",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.38893",
  "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",
      "cond-mat.dis-nn"
    ],
    [
      "arxiv",
      "math.PR"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.38893",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/284bd0nxkxa1m.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:23Z",
      "native_id": "2610.38893",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.38893"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.dis-nn",
      "subfield": "condensed-matter"
    }
  ],
  "title": "Universality in randomly connected neural networks: old and new on a conjecture of Geman and Hwang",
  "title_text": "Universality in randomly connected neural networks: old and new on a conjecture of Geman and Hwang",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/284bd0nxkxa1m",
  "venue": "arXiv",
  "versions": []
}