Resumen de Local single ring theorem
Florent Benaych-Georges
The single ring theorem, by Guionnet, Krishnapur and Zeitouni in Ann. of Math. (2) 174 (2011) 1189–1217, describes the empirical eigenvalue distribution of a large generic matrix with prescribed singular values, that is, an N×NN×N matrix of the form A=UTVA=UTV, with U,VU,V some independent Haar-distributed unitary matrices and TT 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 (logN)−1/4(logN)−1/4. On our way to prove it, we prove a matrix subordination result for singular values of sums of non-Hermitian matrices, as Kargin did in Ann. Probab. 43 (2015) 2119–2150 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.
