{
  "abstract": "We prove quasi-multiplicativity for critical level-sets of Gaussian free fields (GFF) on the metric graphs $$\\widetilde{\\mathbb {Z}}^d$$ Z ~ d ( $$d\\ge 3$$ d ≥ 3 ). Specifically, we study the probability of connecting two general sets located on opposite sides of an annulus with inner and outer radii both of order N , where additional constraints are imposed on the distance of each set to the annulus. We show that for all $$d \\ge 3$$ d ≥ 3 except the critical dimension $$d=6$$ d = 6 , this probability is of the same order as $$N^{(6-d)\\wedge 0}$$ N ( 6 - d ) ∧ 0 (serving as a correction factor) times the product of the two probabilities of connecting each set to the closer boundary of this annulus. The analogue for $$d=6$$ d = 6 is also derived, although the upper and lower bounds differ by a divergent factor of $$N^{o(1)}$$ N o ( 1 ) . Notably, it was conjectured by Basu and Sapozhnikov [3] that quasi-multiplicativity without correction factor holds for Bernoulli percolation on $$\\mathbb {Z}^d$$ Z d when $$3\\le d<6$$ 3 ≤ d < 6 and fails when $$d>6$$ d > 6 . In high dimensions (i.e., $$d>6$$ d > 6 ), taking into account the similarity between the metric graph GFF and Bernoulli percolation (which was proposed by Werner [19] and later partly confirmed by the authors (2024)), our result provides support to the conjecture that Bernoulli percolation exhibits quasi-multiplicativity with correction factor $$N^{6-d}$$ N 6 - d . A crucial application of quasi-multiplicativity is proving the existence of the incipient infinite cluster (IIC), which has been completed by Basu and Sapozhnikov [3] for Bernoulli percolation for $$3\\le d<6$$ 3 ≤ d <",
  "abstract_shown": true,
  "affiliations": [
    "New Cornerstone Science Laboratory, School of Mathematical Sciences, Peking University, No. 5 Yiheyuan Road, Beijing, 100871, China",
    "Faculty of Mathematics and Computer Science, Weizmann Institute of Science, No. 234 Herzl Street, 7610001, Rehovot, Israel"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 2,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Jian Ding"
    },
    {
      "affs": [
        1
      ],
      "name": "Zhenhao Cai"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0t7ce7wv28nnz.bib",
  "cited_by": 12,
  "doi": "10.1007/s00220-026-05772-5",
  "fulltext_url": null,
  "id": "0t7ce7wv28nnz",
  "issns": [
    "0010-3616",
    "1432-0916"
  ],
  "landing_url": "https://doi.org/10.1007/s00220-026-05772-5",
  "language": "en",
  "licence": "CC-BY",
  "licence_label": "CC BY",
  "licence_source": "journal",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "openalex",
      "2610|Stochastic processes and statistical mechanics"
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  "openalex": "W4405253440",
  "pdf_url": "https://link.springer.com/content/pdf/10.1007/s00220-026-05772-5.pdf",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-28",
  "publisher": "Springer Science+Business Media",
  "ris": "https://research.draconitic.com/paper/0t7ce7wv28nnz.ris",
  "sources": [
    {
      "first_seen": "2026-11-08T13:20:36Z",
      "native_id": "W4405253440",
      "source": "openalex",
      "url": "https://doi.org/10.1007/s00220-026-05772-5"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2610|Stochastic processes and statistical mechanics",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Quasi-multiplicativity and Regularity for Metric Graph Gaussian Free Fields",
  "title_text": "Quasi-multiplicativity and Regularity for Metric Graph Gaussian Free Fields",
  "type": "peer-reviewed",
  "updated_at": "2026-11-09T13:21:00Z",
  "url": "https://research.draconitic.com/paper/0t7ce7wv28nnz",
  "venue": "Communications in Mathematical Physics",
  "versions": []
}