On $*$-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 $*$-probability space $(\mathcal{M}_n(\mathbb{C}), \frac{1}{n}\mathrm{tr})$, converges in $*$-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 (\`{a} 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 $*$-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 $n^{-1/2}M_n$.