{
  "abstract": "We prove a functional central limit theorem for trace statistics of Grover walks on random $N$-lifts of a fixed finite connected simple $d$-regular graph, $d\\geq3$. All observations are computed from the same random lift. The squared centred traces, divided by their observation times, have averages converging in probability to a degree-dependent constant. Their partial sums, centred at this constant and divided by the square root of the window length, converge to Brownian motion. For a window of length $M$ beginning at $t\\geq1$, it suffices that $M\\to\\infty$ and $R^4(d-1)^R/N\\to0$, where $R=t+M$. The initial time may remain fixed, with no further restriction on $M/t$. Convergence holds jointly with fixed low-degree polynomial trace statistics; their limiting vector is independent of the Brownian motion. For non-bipartite bases, we also identify Gaussian trace limits at fixed time offsets. Under common initial-time normalisation, subtracting a deterministic growth profile gives an explicit Brownian time change when $M/t$ has a finite limit. Bipartite bases retain the same averaged constant and have twice the Brownian variance rate. The proof combines a growing-dimensional Poisson approximation, a uniform bilinear Chebyshev-energy estimate, a time-decaying Poisson--Gaussian coupling, and a Gaussian quadratic-array theorem. Separate moment estimates give expectation-centred versions under an additional sufficient condition.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.36320",
  "author_total": 2,
  "authors": [
    {
      "name": "Nino Nagata"
    },
    {
      "name": "Ryoichi Suzuki"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d5rdap25k7ddz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.36320",
  "fulltext_url": null,
  "id": "d5rdap25k7ddz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.36320",
  "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|Stochastic processes and statistical mechanics"
    ]
  ],
  "openalex": "W7224116184",
  "pdf_url": "https://arxiv.org/pdf/2610.36320",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d5rdap25k7ddz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:50Z",
      "native_id": "2610.36320",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.36320"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Growing-time trace fluctuations and self-averaging form factors of Grover walks on random lifts",
  "title_text": "Growing-time trace fluctuations and self-averaging form factors of Grover walks on random lifts",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/d5rdap25k7ddz",
  "venue": "arXiv",
  "versions": []
}