{"paper":{"title":"The top eigenvalue of uniformly random trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"G\\'abor Lugosi, Louigi Addario-Berry, Roberto Imbuzeiro Oliveira","submitted_at":"2024-03-13T11:55:57Z","abstract_excerpt":"Let ${\\mathbf T}_n$ be a uniformly random tree with vertex set $[n]=\\{1,\\ldots,n\\}$, let $\\Delta_{{\\mathbf T}_n}$ be the largest vertex degree in ${\\mathbf T}_n$, and let $\\lambda_1({\\mathbf T}_n),\\ldots,\\lambda_n({\\mathbf T}_n)$ be the eigenvalues of its adjacency matrix, arranged in decreasing order. We prove that $|\\lambda_1({\\mathbf T}_n)-\\sqrt{\\Delta_{{\\mathbf T}_n}}| \\to 0$ in expectation as $n \\to \\infty$, and additionally prove probability tail bounds for $|\\lambda_1({\\mathbf T}_n)-\\sqrt{\\Delta_{{\\mathbf T}_n}}|$. Writing $a_n$ for any median of $\\Delta_{{\\mathbf T}_n}$, we also prove "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2403.08443","kind":"arxiv","version":2},"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/2403.08443/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"}