{"paper":{"title":"Universal components of random nodal sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GT","math.PR"],"primary_cat":"math.SP","authors_text":"Damien Gayet (IF), Jean-Yves Welschinger (ICJ)","submitted_at":"2015-03-05T09:27:56Z","abstract_excerpt":"We give, as $L$ grows to infinity, an explicit lower bound of order $L^{n/m}$ for the expected Betti numbers of the vanishing locus of a random linear combination of eigenvectors of $P$ with eigenvalues below $L$. Here, $P$ denotes an elliptic self-adjoint pseudo-differential operator of order $m\\textgreater{}0$, bounded from below and acting on the sections of a Riemannian line bundle over a smooth closed $n$-dimensional manifold  $M$ equipped with some Lebesgue measure. In fact, for every closed hypersurface $\\Sigma$ of $\\mathbb R^n$, we prove that there exists a positive constant $p\\_\\Sigma"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.01582","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"}