{"paper":{"title":"Two conjectures about spectral density of diluted sparse Bernoulli random matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cond-mat.stat-mech","authors_text":"S.K. Nechaev","submitted_at":"2014-09-26T17:53:05Z","abstract_excerpt":"We consider the ensemble of $N\\times N$ ($N\\gg 1$) symmetric random matrices with the bimodal independent distribution of matrix elements: each element could be either \"1\" with the probability $p$, or \"0\" otherwise. We pay attention to the \"diluted\" sparse regime, taking $p=1/N +\\epsilon$, where $0<\\epsilon \\ll 1/N$. In this limit the eigenvalue density, $\\rho(\\lambda)$, is essentially singular, consisting of a hierarchical ultrametric set of peaks. We provide two conjectures concerning the structure of $\\rho(\\lambda)$: (i) we propose an equation for the position of sequential (in heights) pea"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.7650","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"}