{
  "abstract": "We study the asynchronous $\\ell^p$ relaxation introduced by Amir, Nazarov, and Peres: at each step, a uniformly chosen vertex minimizes its incident $\\ell^p$ energy. For the profile $f_t$ after $t$ updates, let $$ \\mathsf T_p(G,1/2):= \\sup_{\\lVert f_0\\rVert_\\infty\\le1} \\mathbb{E}\\bigl[\\min\\{t\\ge0:\\operatorname{osc}(f_t)\\le1/2\\}\\bigr]. $$ For $1<p<\\infty$, we obtain graph-dependent estimates for boxes, trees, and conductance expanders. On the nearest-neighbor box $[L]^d$, for $d,L\\ge2$ and $n=L^d$, the answer is, up to logarithmic factors, $nd^{1/(p-1)}L^{p/(p-1)}$ for $1<p<2$ and $ndL^2$ for $p\\ge2$. On bounded-degree trees, an explicit rerooting-invariant parameter $T_G$ determines the answer up to logarithmic factors; for arbitrary trees it gives upper and lower bounds that differ additionally by the maximum degree. If the volume conductance $h(G)\\ge h_0>0$, then $\\mathsf T_p(G,1/2)=Θ_{p,h_0}(n\\log n)$ without a degree assumption. At $p=\\infty$, every connected graph satisfies $\\mathsf T_\\infty(G,1/2)\\ge c nD^2/Δ$, where $D$ and $Δ$ are its diameter and maximum degree.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.03856",
  "author_total": 1,
  "authors": [
    {
      "name": "Chenyu Gan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4fy2ag12w1zxd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.03856",
  "fulltext_url": null,
  "id": "4fy2ag12w1zxd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.03856",
  "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",
      "math.PR"
    ],
    [
      "openalex",
      "2610|Stochastic processes and statistical mechanics"
    ]
  ],
  "openalex": "W7208744723",
  "pdf_url": "https://arxiv.org/pdf/2609.03856",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4fy2ag12w1zxd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:36Z",
      "native_id": "2609.03856",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.03856"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Consensus time for asynchronous $\\ell^p$ relaxation: graph dependence",
  "title_text": "Consensus time for asynchronous ℓ^p relaxation: graph dependence",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:36Z",
  "url": "https://research.draconitic.com/paper/4fy2ag12w1zxd",
  "venue": "arXiv",
  "versions": []
}