On $\ast$-Convergence of Schur-Hadamard Products of Independent Nonsymmetric Random Matrices

Abstract

Let $\{x_{\alpha }{\}}_{\alpha \in \mathbb{Z}}$ and $\{y_{\alpha }{\}}_{\alpha \in \mathbb{Z}}$ be two independent collections of zero mean, unit variance random variables with uniformly bounded moments of all orders. Consider a nonsymmetric Toeplitz matrix $X_n=((x_{i-j}){)}_{1\le i,j\le n}$ and a Hankel matrix $Y_n=((y_{i+j}){)}_{1\le i,j\le n}$, and let $M_n=X_n\odot Y_n$ be their elementwise/Schur-Hadamard product. In this article, we show that almost surely, $n^{-1/2}{M}_n$, as an element of the $\ast$-probability space $({\mathcal{M}}_n(\mathbb{C}),\frac{1}{n}\mathrm{tr})$, converges in $\ast$-distribution to a circular variable. With i.i.d. Rademacher entries, this construction gives a matrix model for circular variables with only $O(n)$ bits of randomness. We also consider a dependent setup where $\{x_{\alpha }\}$ and $\{y_{\beta }\}$ are independent strongly multiplicative systems (à la Gaposhkin [7]) satisfying an additional \emph{admissibility} condition, and have uniformly bounded moments of all orders -- a nontrivial example of such a system being $\{\sqrt{2}\sin (2^n\pi U){\}}_{n\in {\mathbb{Z}}_{+}}$, where $U\sim \mathrm{Uniform}(0,1)$. In this case, we show in-expectation and in-probability convergence of the $\ast$-moments of $n^{-1/2}{M}_n$ to those of a circular variable. Finally, we generalise our results to Schur-Hadamard products of structured random matrices of the form $X_n=((x_{L_X(i,j)}){)}_{1\le i,j\le n}$ and $Y_n=((y_{L_Y(i,j)}){)}_{1\le i,j\le n}$, under certain assumptions on the \emph{link-functions} $L_X$ and $L_Y$, most notably the injectivity of the map $(i,j)\mapsto (L_X(i,j),L_Y(i,j))$. Based on numerical evidence, we conjecture that the circular law ${\mu }_{\mathrm{circ}}$, i.e. the uniform measure on the unit disk of $\mathbb{C}$, which is also the Brown measure of a circular variable, is in fact the limiting spectral measure of $n^{-1/2}{M}_n$.

18 pages, 2 figures, to appear in IMRN

References in corpus (3)

• Spectral measure of large random Hankel, Markov and Toeplitz matrices
• Eigenvalue Spacing Distribution for the Ensemble of Real Symmetric Toeplitz Matrices
• Some results on random circulant matrices