Local Single Ring Theorem

Understand

The Single Ring Theorem, by Guionnet, Krishnapur and Zeitouni, describes the empirical eigenvalue distribution of a large generic matrix with prescribed singular values, i.e.

• 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 Kargin did for Hermitian matrices.