Colloquium: Random Commuting Matrices

Abstract: The study of the eigenvalue distribution of random matrices is a well-established field, dating back to the 1920’s. It became popular with the work of Wigner and Dyson in the 1950’s and 60’s, and today is a major field in both probability and theoretical physics. Typically a random matrix is generated by choosing the entries identically and independently distributed. What can one say about random d-tuples of commuting matrices? What does it even mean, since one can no longer choose entries independently? We will describe an approach to defining a random d-tuple of commuting matrices. We will show that in the Hermitian case, the description of their eigenvalue distribution parallels to some extent the single matrix theory, though there is a qualitative change when d ≥ 5. In the non-self adjoint case the eigenvalue distribution is quite unlike the single matrix case.