Spectral distributions of periodic random matrix ensembles

Abstract

Kologlu, Kopp and Miller compute the limiting spectral distribution of a certain class of real random matrix ensembles, known as k-block circulant ensembles, and discover that it is exactly equal to the eigenvalue distribution of an k × k Gaussian unitary ensemble. We give a simpler proof that under very general conditions which subsume the cases studied by Kologlu-Kopp-Miller, real-symmetric ensembles with periodic diagonals always have limiting spectral distribution equal to the eigenvalue distribution of a finite Hermitian ensemble with Gaussian entries which is a 'complex version' of a k × k submatrix of the ensemble. We also prove an essentially algebraic relation between certain periodic finite Hermitian ensembles with Gaussian entries, and the previous result may be seen as an asymptotic version of this for real-symmetric ensembles. The proofs show that this general correspondence between periodic random matrix ensembles and finite complex Hermitian ensembles is elementary and combinatorial in nature.

Limits and fluctuations of p-adic random matrix products

“…The techniques we develop here also apply, with some additional work, in the problem of the limiting distribution of nilpotent Jordan blocks (equivalently, ranks of powers) of a random n×n strictly upper-triangular matrix A over F q , studied by Kirillov [53] and Borodin [14,15]. The same random variable L t,χ , with t = q −1 and χ depending on the subsequence along which n is sent to ∞, also appears in these limits [74].…”

“…) which will in general not be a nonnegative real number. We will not need the case of general complex τ in this work, but show it because it will be useful in a subsequent one [74] and the proof is essentially the same as in the case of positive real τ when the above expression is a bona fide probability.…”