{"paper":{"title":"Condensation of the roots of real random polynomials on the real axis","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cond-mat.stat-mech","math.MP"],"primary_cat":"math-ph","authors_text":"Gregory Schehr, Satya N. Majumdar","submitted_at":"2009-02-06T09:21:43Z","abstract_excerpt":"We introduce a family of real random polynomials of degree n whose coefficients a_k are symmetric independent Gaussian variables with variance <a_k^2> = e^{-k^\\alpha}, indexed by a real \\alpha \\geq 0. We compute exactly the mean number of real roots <N_n> for large n. As \\alpha is varied, one finds three different phases. First, for 0 \\leq \\alpha < 1, one finds that <N_n> \\sim (\\frac{2}{\\pi}) \\log{n}. For 1 < \\alpha < 2, there is an intermediate phase where < N_n > grows algebraically with a continuously varying exponent, < N_n > \\sim \\frac{2}{\\pi} \\sqrt{\\frac{\\alpha-1}{\\alpha}} n^{\\alpha/2}. "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0902.1027","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"}