Real eigenvalues of non-Gaussian random matrices and their products

Random right eigenvalues of Gaussian quaternionic matrices

We consider a random matrix whose entries are independent Gaussian variables taking values in the field of quaternions with variance 1/n. Using logarithmic potential theory, we prove the almost sure convergence, as the dimension n goes to infinity, of the empirical distribution of the right eigenvalues towards some measure supported on the unit ball of the quaternions field. Some comments on mo...

Extreme value statistics of eigenvalues of Gaussian random matrices.

We compute exact asymptotic results for the probability of the occurrence of large deviations of the largest (smallest) eigenvalue of random matrices belonging to the Gaussian orthogonal, unitary, and symplectic ensembles. In particular, we show that the probability that all the eigenvalues of an (NxN) random matrix are positive (negative) decreases for large N as approximately exp [-beta theta...

Gaussian Fluctuations of Eigenvalues in Wigner Random Matrices

Abstract. We study the fluctuations of eigenvalues from a class of Wigner random matrices that generalize the Gaussian orthogonal ensemble. We begin by considering an n × n matrix from the Gaussian orthogonal ensemble (GOE) or Gaussian symplectic ensemble (GSE) and let xk denote eigenvalue number k. Under the condition that both k and n − k tend to infinity as n → ∞, we show that xk is normally...

Perturbing non-real eigenvalues of nonnegative real matrices

Products of independent non-Hermitian random matrices

We consider the product of a finite number of non-Hermitian random matrices with i.i.d. centered entries of growing size. We assume that the entries have a finite moment of order bigger than two. We show that the empirical spectral distribution of the properly normalized product converges, almost surely, to a non-random, rotationally invariant distribution with compact support in the complex pl...