{"paper":{"title":"Index statistical properties of sparse random graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cond-mat.dis-nn","math-ph","math.MP"],"primary_cat":"cond-mat.stat-mech","authors_text":"Daniel A. Stariolo, Fernando L. Metz","submitted_at":"2015-09-04T21:29:50Z","abstract_excerpt":"Using the replica method, we develop an analytical approach to compute the characteristic function for the probability $\\mathcal{P}_N(K,\\lambda)$ that a large $N \\times N$ adjacency matrix of sparse random graphs has $K$ eigenvalues below a threshold $\\lambda$. The method allows to determine, in principle, all moments of $\\mathcal{P}_N(K,\\lambda)$, from which the typical sample to sample fluctuations can be fully characterized. For random graph models with localized eigenvectors, we show that the index variance scales linearly with $N \\gg 1$ for $|\\lambda| > 0$, with a model-dependent prefacto"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.01614","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":""},"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"}