{"paper":{"title":"Norm Bounds for Sparse Random Tensors and Spectral Gap of Random Hypergraphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.DM","math.CO"],"primary_cat":"math.PR","authors_text":"Kevin Lucca, Lucas Pesenti","submitted_at":"2026-07-08T11:51:36Z","abstract_excerpt":"Friedman and Wigderson (1995) introduced a notion of second eigenvalue for hypergraphs that generalizes the second eigenvalue of the adjacency matrix of a graph. We show that $r$-uniform Erd\\H{o}s-R\\'enyi hypergraphs on $n$ vertices exhibit a spectral gap as soon as their expected number of hyperedges $m$ satisfies $m \\gg n^{r/2}$. Prior work identified this scale only up to logarithmic factors; removing these factors is the main technical challenge.\n  Our proof overcomes this obstacle through an explicit decomposition of an associated selector process, inspired by a generic decomposition theo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.07308","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/2607.07308/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"}