{
  "abstract": "We analyse the large-time asymptotics of the Schrödinger time evolution of a free quantum particle on the Erdős-Rényi graph $\\mathbb{G}(N,d/N)$, where the quantum Hamiltonian is given by the adjacency matrix of $\\mathbb{G}$. Starting from a state localized at a vertex, the wave function is approximately radial, and we characterize precisely its asymptotic form. Our results in particular cover vertices whose degree is close to $2d$, hence probing the emergence of bound states from a dynamical perspective. This provides a dynamical counterpart of the coexistence of localized and delocalized states established in [arXiv:1905.03243,arXiv:2005.14180,arXiv:2106.12519,arXiv:2109.03227,arXiv:2305.16294], with the former giving rise to a localized mode and the latter to a dispersive wave. Our results hold in the regime $\\sqrt{\\log N} \\ll d \\ll N^{o(1)}$, which is the full region in which such dynamical phenomena may exist, and for times $1 \\ll t \\ll T_N$ for some explicit $T_N \\gg 1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.35867",
  "author_total": 3,
  "authors": [
    {
      "name": "Johannes Alt"
    },
    {
      "name": "Raphael Ducatez"
    },
    {
      "name": "Antti Knowles"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fepgqe6vyywnt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.35867",
  "fulltext_url": null,
  "id": "fepgqe6vyywnt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.35867",
  "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",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7223803180",
  "pdf_url": "https://arxiv.org/pdf/2610.35867",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fepgqe6vyywnt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:35Z",
      "native_id": "2610.35867",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.35867"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Quantum dynamics on the sparse Erdős-Rényi graph",
  "title_text": "Quantum dynamics on the sparse Erdős-Rényi graph",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:43Z",
  "url": "https://research.draconitic.com/paper/fepgqe6vyywnt",
  "venue": "arXiv",
  "versions": []
}