{"paper":{"title":"Concentration of the spectral norm of Erd\\H{o}s-R\\'enyi random graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"G\\'abor Lugosi, Nikita Zhivotovskiy, Shahar Mendelson","submitted_at":"2018-01-07T08:23:15Z","abstract_excerpt":"We present results on the concentration properties of the spectral norm $\\|A_p\\|$ of the adjacency matrix $A_p$ of an Erd\\H{o}s-R\\'enyi random graph $G(n,p)$. First we consider the Erd\\H{o}s-R\\'enyi random graph process and prove that $\\|A_p\\|$ is uniformly concentrated over the range $p\\in [C\\log n/n,1]$. The analysis is based on delocalization arguments, uniform laws of large numbers, together with the entropy method to prove concentration inequalities. As an application of our techniques we prove sharp sub-Gaussian moment inequalities for $\\|A_p\\|$ for all $p\\in [c\\log^3n/n,1]$ that improve"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.02157","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}