{"paper":{"title":"Super-Brownian limits and the $k$-point function for high-dimensional percolation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Arthur Blanc-Renaudie, Tom Hutchcroft","submitted_at":"2026-07-24T15:12:59Z","abstract_excerpt":"We prove that there exist positive constants $A$ and $V$ such that the high-dimensional critical percolation $k$-point function is given by \\[ T_{p_c}(x_1,x_2,\\ldots,x_{k}) \\sim V^{k-2} A^{2k-3} \\sum_{T\\in \\mathsf{Tr}(k)} \\sum_{\\substack{\\Phi:V(T)\\to \\mathbb{Z}^d \\\\ \\Phi(i)=x_i \\forall 1\\leq i \\leq k}} \\prod_{\\substack{u,v\\in V(T)\\\\u\\sim v}}G(\\Phi(u),\\Phi(v)) \\] as $\\min_{i\\neq j}\\|x_i-x_j\\|\\to \\infty$, where $\\mathsf{Tr}(k)$ is a set of isomorphism class representatives of trees with $k$ labelled leaves $\\{1,\\ldots,k\\}$ and unlabelled internal vertices all of which have degree $3$ and $G$ is "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.22387","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/2607.22387/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"}