{"paper":{"title":"On pure complex spectrum for truncations of random orthogonal matrices and Kac polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.FA","math.MP"],"primary_cat":"math.PR","authors_text":"Martin Gebert, Mihail Poplavskyi","submitted_at":"2019-05-08T15:32:24Z","abstract_excerpt":"Let $O(2n+\\ell)$ be the group of orthogonal matrices of size $\\left(2n+\\ell\\right)\\times \\left(2n+\\ell\\right)$ equipped with the probability distribution given by normalized Haar measure. We study the probability \\begin{equation*} p_{2n}^{\\left(\\ell\\right)} = \\mathbb{P}\\left[M_{2n} \\, \\mbox{has no real eigenvalues}\\right], \\end{equation*} where $M_{2n}$ is the $2n\\times 2n$ left top minor of a $(2n+\\ell)\\times(2n+\\ell)$ orthogonal matrix. We prove that this probability is given in terms of a determinant identity minus a weighted Hankel matrix of size $n\\times n$ that depends on the truncation "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.03154","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"}