{
  "abstract": "We introduce a new family of trickledown theorems, a.k.a., local to global technique to bound the spectral gap of the Glauber dynamics for multi-state spin systems. In this technique instead of upper-bounding the influence matrix of a link of co-dimension 2 by $λI$ (where $λ$ is the second eigenvalue of the link), we upper-bound the influence matrix after a carefully chosen rank-1 shift. The rank-1 shift allows for a significantly smaller upper-bound but it comes at the cost of bounding the average loss due to rank-1 perturbations. As an application we use this method to show that the natural Glauber dynamics mixes in polynomial time to generate samples from the Sherrington-Kirkpatrick model for $β\\leq \\tfrac{1}{2}+\\varepsilon$, for an absolute constant $\\varepsilon>0$. At the heart of the proof we manage to bound the loss due to rank-1 perturbations by averaging over all links of co-dimension 2.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.13138",
  "author_total": 3,
  "authors": [
    {
      "name": "Mathews Boban"
    },
    {
      "name": "Anqi Li"
    },
    {
      "name": "Shayan Oveis Gharan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/873949w1jyk4n.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.13138",
  "fulltext_url": null,
  "id": "873949w1jyk4n",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.13138",
  "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",
      "cs.DS"
    ],
    [
      "arxiv",
      "math-ph"
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      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.PR"
    ],
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      "openalex",
      "2613|Markov Chains and Monte Carlo Methods"
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  "openalex": "W7212756553",
  "pdf_url": "https://arxiv.org/pdf/2609.13138",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/873949w1jyk4n.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:07Z",
      "native_id": "2609.13138",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.13138"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Rank-1-perturbed trickledown theorems: Mixing time of Glauber dynamics for the Sherrington-Kirkpatrick model up to $β\\leq \\frac{1}{2}+\\varepsilon$",
  "title_text": "Rank-1-perturbed trickledown theorems: Mixing time of Glauber dynamics for the Sherrington-Kirkpatrick model up to β≤ 1/2+ε",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:11Z",
  "url": "https://research.draconitic.com/paper/873949w1jyk4n",
  "venue": "arXiv",
  "versions": []
}