2017/04/20 by Mihai Popa, Popa, Mihai, Zhiwei Hao +1
Mathematics · #06A07 #46L54 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #FOS: Mathematics #Operator Algebras (math.OA) #Random Matrices and Applications
paper · pdf · doi:10.48550/arxiv.1704.06324
openalex publication_date 2017/04/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The paper gives a general condition on permutations, condition under which a semicircular matrix is free independent, or asymptotically free independent from the semicircular matrix obtained by permuting its entries. In particular, it is shown that semicircular matrices are asymptotically free from their transposes, a result similar to the case of Gaussian random matrices. There is also an analysis of asymptotic second order relations between semicircular matrices and their transposes, with results not very similar to the commutative (i.e. Gaussian random matrices) framework. The paper also presents an application of the main results to the study of Gaussian random matrices and furthermore it is shown that the same condition as in the case of semicircular matrices gives Boolean independence, or asymptotic Boolean independence when applied to Bernoulli matrices.