Limit Laws for Random Matrices from Traffic-Free Probability

Benson Au

Published 2016-01-10Version 1

We consider a random matrix ensemble that interpolates between the Wigner matrices and the random Markov matrices studied by Bryc, Dembo, and Jiang. For a real Wigner matrix $\mathbf{W}_n$, let $\mathbf{D}_n$ be the diagonal matrix whose entries correspond to the row sums of $\mathbf{W}_n$, i.e., $\mathbf{D}_n(i,i) = \sum_{j=1}^n \mathbf{W}_n(i,j)$. For $p,q\in\mathbb{R}$, we study the asymptotic behavior of the matrices $\mathbf{M}_{n,p,q}=p\mathbf{W}_n+q\mathbf{D}_n$ as $n\to\infty$. By realizing $\mathbf{M}_{n,p,q}$ as a traffic of $\mathbf{W}_n$ in the sense of Male, we show that when the entries of $\mathbf{W}_n$ have finite moments of all orders, independent $\mathbf{M}_{n,p,q}$ are asymptotically traffic-free with stable universal limiting traffic distribution. We obtain the limiting spectral distributions of the ensemble $(\mathbf{M}_{n,p,q})_{p,q\in\mathbb{R}}$ as a consequence via the traffic-free central limit theorem. The continuity of the limiting spectral distribution in the parameters $p,q$ allows us to incorporate additional randomness: if $(X_n)$ and $(Y_n)$ are sequences of real-valued random variables converging a.s. to $X$ and $Y$ respectively, we show that the empirical spectral distributions $\mu(\mathbf{M}_{n,X_n,Y_n})$ converge weakly a.s. to the random free convolution $\mathcal{SC}(0,X^2)\boxplus\mathcal{N}(0,Y^2)$.