Fluctuations and large deviations of some perturbed random matrices

Summary

We review joint results with Benaych-Georges and Guionnet (Electron. J. Probab. 16:60 (2011), 1621–1662 and Prob. Theory Rel. Fields 154:3–4 (2012), 703–751) about fluctuations and large deviations of some spiked models, putting them in the perspective of various works of the last years on extreme eigenvalues of finite-rank deformations of random matrices.

1.Introduction

General statement of the problem. The following algebraic problem is very classical: ,Let A and B be two Hermitian matrices of the same size. Assume we know the spectrum of each of them. What can be said about the spectrum of their sum A + B? The problem was posed by Weyl [1912]. He gave a series of necessary conditions, known as Weyl's interlacing inequalities: if λ₁(A)≥...≥ λ_n(A), λ₁(B)≥...≥ λ_n(B) and λ₁(A + B) ≥...≥λ_n(A + B) are the spectra of A, B and A + B, then

whenever 0≤i; j; i + j <n. These inequalities have been very fruitful in various fields.

After that, it took a long time to get necessary and sufficient conditions. Horn in the sixties formulated the right conjecture, but the final answer was only given in the late nineties in a series a papers [Klyachko 1998; Helmke and Rosenthal 1995; Knutson and Tao 2001].

If we now look at the problem asymptotically, namely when the size of both matrices goes to infinity, an important breakthrough was made by free probability theory with the notion of asymptotic freeness. This property can be roughly stated as follows: If A and B are large-dimensional and in generic position relative to one another, the limiting spectrum of their sum depends only on their respective spectra and is given by the free convolution of the two spectra. We won't go

further into free probability theory, but we refer the reader to [Emery et al. 2000] for general background on free probability.