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 more general Gaussian quaternionic random matrix models are also made. Introduction Our motivation for studying quaternionic random matrices comes from the following facts. The projection onto the complex plane of the uniform measure on the unit sphere S3 of R4 is the uniform measure on the unit disk D(0, 1) of C, also called the circular law. Furthermore, the projection onto the real axis of the uniform measure on D(0, 1) is the semi-circular law on [−1, 1]. The last two measures play a key role in random matrix theory. Indeed, it is well known since Wigner’s paper [12] that as the dimension goes to infinity, the empirical distribution of the eigenvalues of a Gaussian Hermitian random matrix converges to the semi-circular law, and it has also been proved that the empirical distribution of the eigenvalues of a complex Gaussian random matrix converges to the circular law (see e.g. the book of Mehta [9] or the paper of Tao, Vu and Krishnapur [11]). Our initial idea, due to Philippe Biane, was to find a random matrix model such that the empirical spectral measure would converge to the uniform measure on the unit sphere S3, and thus, in view of the previous observations, to study quaternionic random matrices, the unit sphere of quaternions being naturally identified with the unit sphere S3. The hope was then, after defining properly the eigenvalues of quaternionic random matrices, that the empirical spectral distribution of a quaternionic Gaussian matrix will converge to the uniform distribution on the unit sphere of quaternions. We will see that in fact, this is not the case, and we will prove a convergence result for the empirical spectral distribution towards some measure supported by the unit ball of the quaternions field. The paper is organized as follows. In Section 1, we recall some basic facts on the quaternions field H and on matrices of quaternions. Note that due Date: September 2, 2011. 2000 Mathematics Subject Classification. 15A52, 60B15.