Circular law for random matrices with exchangeable entries

Radosław Adamczak (1) , Djalil Chafaï (2, 3) , Paweł Wolff (1)

1 Institute of Mathematics [Warsaw]
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
3 PSL - Université Paris Sciences et Lettres

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.

Mots clés: Random permutations; Random matrices; Combinatorial Central Limit Theorem; exchangeable distributions; concentration of measure; smallest singular value; spectral analysis

Domaines: Probabilités [math.PR]

Radosław Adamczak, Djalil Chafaï, Paweł Wolff. Circular law for random matrices with exchangeable entries. Random Structures and Algorithms, 2016, 48 (3), pp.454-479. ⟨10.1002/rsa.20599⟩. ⟨hal-00947145⟩