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Local single ring theorem. (English) Zbl 1383.15033
Summary: The single ring theorem, by A. Guionnet et al. [Ann. Math. (2) 174, No. 2, 1189–1217 (2011; Zbl 1239.15018)], describes the empirical eigenvalue distribution of a large generic matrix with prescribed singular values, that is, an \(N\times N\) matrix of the form \(A=UTV\), with \(U\), \(V\) some independent Haar-distributed unitary matrices and \(T\) a deterministic matrix whose singular values are the ones prescribed. In this text, we give a local version of this result, proving that it remains true at the microscopic scale \((\log N)^{-1/4}\). On our way to prove it, we prove a matrix subordination result for singular values of sums of non-Hermitian matrices, as V. Kargin did [Ann. Probab. 43, No. 4, 2119–2150 (2015; Zbl 1320.60022)] for Hermitian matrices. This allows to prove a local law for the singular values of the sum of two non-Hermitian matrices and a delocalization result for singular vectors.

MSC:
15B52 Random matrices (algebraic aspects)
15A18 Eigenvalues, singular values, and eigenvectors
60B20 Random matrices (probabilistic aspects)
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