Eigenvectors and eigenvalues in a random subspace of a tensor product

• 成果类型: Article
• 署名作者: Belinschi, Serban; Collins, Benoit; Nechita, Ion
• 署名单位: University of Ottawa; University of Saskatchewan; Centre National de la Recherche Scientifique (CNRS); Ecole Centrale de Lyon; Institut National des Sciences Appliquees de Lyon - INSA Lyon; Universite Claude Bernard Lyon 1; Universite Jean Monnet; Institute of Mathematics of the Romanian Academy; Romanian Academy
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-012-0386-3
• 发表日期: 2012
• 页码: 647-697
• 关键词: free convolution; random-variables; REGULARITY; entropy

摘要

Given two positive integers n and k and a parameter ta(0,1), we choose at random a vector subspace V (n) aS,a, (k) aSua, (n) of dimension N similar to 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 ae (k) . Our proof relies on free probability, random matrix theory, complex analysis and matrix analysis techniques. The main result comes together with a law of large numbers for the singular value decomposition of the eigenvectors corresponding to large eigenvalues of a random truncation of a matrix with high eigenvalue degeneracy.