2014/02/15 by Radosław Adamczak, Djalil Chafaï, Paweł Wolff · 39 citations
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Circular law #Combinatorics #Discrete mathematics #Eigenvalues and eigenvectors #Hermitian matrix #Mathematics #Matrix (chemical analysis) #Multivariate random variable #Physics #Pure mathematics #Quantum mechanics #Random Matrices and Applications #Random field #Random function #Random matrix #Random permutation #Random variable #Singular value #Statistics #Sum of normally distributed random variables #Symmetric group #Universality (dynamical systems) #math.PR
paper · pdf · doi:10.1002/rsa.20599
published in Random Structures and Algorithms 48(3), 454-479 (Wiley)
arxiv created 2014/02/15 · openalex publication_date 2015/09/08 · arxiv updated 2016/03/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Abstract An exchangeable random matrix is a random matrix with distribution invariant under any permutation of the entries. For such random matrices, we show, as the dimension tends to infinity, that the empirical spectral distribution tends to the uniform law on the unit disc. This is an instance of the universality phenomenon known as the circular law, for a model of random matrices with dependent entries, rows, and columns. It is also a non‐Hermitian counterpart of a result of Chatterjee on the semi‐circular law for random Hermitian matrices with exchangeable entries. The proof relies in particular on a reduction to a simpler model given by a random shuffle of a rigid deterministic matrix, on hermitization, and also on combinatorial concentration of measure and combinatorial Central Limit Theorem. A crucial step is a polynomial bound on the smallest singular value of exchangeable random matrices, which may be of independent interest. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 454–479, 2016