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Circular law for random matrices with exchangeable entries

Adamczak, R.; Chafaï, Djalil; Wolff, P. (2016), Circular law for random matrices with exchangeable entries, Random Structures & Algorithms, 48, 3, p. 454-479. 10.1002/rsa.20599

Type
Article accepté pour publication ou publié
External document link
https://arxiv.org/abs/1402.3660v1
Date
2016
Journal name
Random Structures & Algorithms
Volume
48
Number
3
Publisher
Wiley
Published in
Paris
Pages
454-479
Publication identifier
10.1002/rsa.20599
Metadata
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Author(s)
Adamczak, R.
Chafaï, Djalil cc
Wolff, P.
Abstract (EN)
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.
Subjects / Keywords
smallest singular value; spectral analysis; Random permutations; exchangeable distributions; concentration of measure; Combinatorial Central Limit Theorem; Random matrices

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