{"paper":{"title":"Cluster volumes for the Gaussian free field on metric graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Alexander Drewitz, Alexis Pr\\'evost, Pierre-Fran\\c{c}ois Rodriguez","submitted_at":"2024-12-09T18:56:45Z","abstract_excerpt":"We study the volume of the critical clusters for the percolation of the level sets of the Gaussian free field on metric graphs. On $\\mathbb{Z}^d$ below the upper-critical dimension $d=6$, we show that the largest such cluster in a box of side length $r$ has volume of order $r^{\\frac{d+2}{2}}$, as conjectured by Werner in arXiv:2002.11487. This is in contrast to the mean-field regime $d>6$, where this volume is of order $r^4$. We further obtain precise asymptotic tails for the volume of the critical cluster of the origin, and a lower bound on the tail of the volume of the near-critical cluster "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.06772","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2412.06772/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}