{
  "abstract": "Let $D_n$ be the dihedral group of order $2n$. Consider a continuous-time random walk on $D_n$ driven by arbitrary symmetric rates whose support generates $D_n$. For $p\\in[1,\\infty]$, we say the pair $(D_n,p)$ is rate-monotonic if for each fixed time $t$, the $\\ell^p$-distance between the random walk's distribution at time $t$ and the uniform distribution is monotonically decreasing as a function of the rates. Lyons and White proved that $(D_n,2)$ and $(D_n,\\infty)$ are rate-monotonic. Somewhat counterintuitively, they found several pairs $(D_n,p)$ with ${p\\in[1,1.997]\\cup[2.001,3.999]\\cup[4.001,5.995]}$ that are not rate-monotonic, and they asked whether any such pairs exist with $p=4$ or $p=6$. We resolve their question, proving that $(D_n,2m)$ is rate-monotonic for all positive integers $m$ and $n$. In fact, we prove a generalization of this result to a broader family of groups that includes generalized dihedral groups, dicyclic groups, and generalized quaternion groups. In the other direction, we prove that for every real $p\\geq 1$ that is not an even integer, there exists a positive integer $n$ such that $(D_n,p)$ is not rate-monotonic. The results of this paper were formally verified in Lean by AxiomProver assuming standard literature.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.27708",
  "author_total": 2,
  "authors": [
    {
      "name": "Colin Defant"
    },
    {
      "name": "Ken Ono"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d6xejve006m1k.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.27708",
  "fulltext_url": null,
  "id": "d6xejve006m1k",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.27708",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2610|Stochastic processes and statistical mechanics"
    ]
  ],
  "openalex": "W7204809563",
  "pdf_url": "https://arxiv.org/pdf/2608.27708",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d6xejve006m1k.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:28Z",
      "native_id": "2608.27708",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.27708"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Proof of the Lyons--White Conjecture",
  "title_text": "Proof of the Lyons--White Conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:28Z",
  "url": "https://research.draconitic.com/paper/d6xejve006m1k",
  "venue": "arXiv",
  "versions": []
}