Circular law for random matrices with exchangeable entries

Radosław Adamczak,Paweł Wolff,Djalil Chafaï

doi:10.1002/rsa.20599

Abstract

AbstractAn 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

Highlights

An exchangeable random matrix is a random matrix with distribution invariant under any permutation of the entries
We show that the asymptotic behavior of the spectral measure is governed by the circular law
The precise formulation is given in Theorem 1.2 below. This first result is used to deduce the circular law for a larger class of random matrices with exchangeable entries, as stated in Theorem 1.3 below. This second result can be seen as a counterpart of the one of Chatterjee [13], who proved that the spectral measure of symmetric random matrices with exchangeable entries in the upper triangle converges to Wigner’s semi-circular law, which is the universal limit for random matrices with i.i.d. entries of finite variance

Summary

Circular law for random matrices with exchangeable entries

To cite this version: Radoslaw Adamczak, Djalil Chafaï, Pawel Wolff. Circular law for random matrices with exchangeable entries. Random Structures and Algorithms, Wiley, 2016, 48 (3), pp.454479. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés

Var W

Now let

The function t

Taking another union bound over all i