{"paper":{"title":"Laws of large numbers for eigenvectors and eigenvalues associated to random subspaces in a tensor product","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OA","quant-ph"],"primary_cat":"math.PR","authors_text":"B. Collins, I. Nechita, S. Belinschi","submitted_at":"2010-08-18T14:54:26Z","abstract_excerpt":"Given two positive integers $n$ and $k$ and a parameter $t\\in (0,1)$, we choose at random a vector subspace $V_{n}\\subset \\mathbb{C}^{k}\\otimes\\mathbb{C}^{n}$ of dimension $N\\sim tnk$. We show that the set of $k$-tuples of singular values of all unit vectors in $V_n$ fills asymptotically (as $n$ tends to infinity) a deterministic convex set $K_{k,t}$ that we describe using a new norm in $\\R^k$.\n  Our proof relies on free probability, random matrix theory, complex analysis and matrix analysis techniques. The main result result comes together with a law of large numbers for the singular value de"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.3099","kind":"arxiv","version":3},"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"}