Abstract
These expository notes are centered around the circular law theorem, which states that the empirical spectral distribution of a n×n random matrix with i.i.d. entries of variance 1/n tends to the uniform law on the unit disc of the complex plane as the dimension n tends to infinity. This phenomenon is the non-Hermitian counterpart of the semi circular limit for Wigner random Hermitian matrices, and the quarter circular limit for Marchenko-Pastur random covariance matrices. We present a proof in a Gaussian case, due to Silverstein, based on a formula by Ginibre, and a proof of the universal case by revisiting the approach of Tao and Vu, based on the Hermitization of Girko, the logarithmic potential, and the control of the small singular values. Beyond the finite variance model, we also consider the case where the entries have heavy tails, by using the objective method of Aldous and Steele borrowed from randomized combinatorial optimization. The limiting law is then no longer the circular law and is related to the Poisson weighted infinite tree. We provide a weak control of the smallest singular value under weak assumptions, using asymptotic geometric analysis tools. We also develop a quaternionic Cauchy-Stieltjes transform borrowed from the Physics literature.
Highlights
AMS 2000 subject classifications: Primary 15B52; secondary 60B20, 60F15
The notes end with appendix A devoted to a novel general weak control of the smallest singular value of random matrices with i.i.d. entries, with weak assumptions, well suited for the proof of the circular law theorem and its heavy tailed analogue
The quaternionic Ginibre Ensemble was considered at the origin by Ginibre [53]. It has been recently shown [18] by using the logarithmic potential that there exists an analogue of the circular law theorem for this ensemble, in which the limiting law is supported in the unit ball of the quaternions field
Summary
The eigenvalues of A ∈ Mn(C) are the roots in C of its characteristic polynomial PA(z) := det(A − zI). The relationships between the eigenvalues and the singular values are captured by a set of inequalities due to Weyl [153]2, which can be obtained by using the Schur unitary triangularization, see for instance [82, Theorem 3.3.2 page 171]. It turns out that for non-normal matrices, the eigenvalues are more sensitive to perturbations than the singular values. 1, κ2n) we get s1(A) = · · · = sn−1(A) = 1, sn(A) = 0 and s1(B) = · · · = sn−1(B) = 1, sn(B) = κn and for any choice of κn, since the atom κn has weight 1/n, νA δ1 and νB δ1 This example shows the stability of the limiting distribution of singular values under an additive perturbation of rank 1 of arbitrary large norm, and the instability of the limiting eigenvalues distribution under an additive perturbation of rank 1 of arbitrary small norm (κ1n/n → 0). See [150] for more
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