On *-Convergence of Schur–Hadamard Products of Independent Nonsymmetric Random Matrices

Abstract

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 $(\mathcal {M}_n({\mathbb {C}}), \frac {1}{n}\text {tr})$, 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 (à la Gaposhkin [7]) satisfying an additional 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 $U \sim \textrm {Uniform}(0, 1)$. 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 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 $\mu _{\textrm {circ}}$, that is, 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 (LSM)of $n^{-1/2}M_n$. If true, this would furnish an interesting example where a random matrix with only $O(n)$ bits of randomness has the circular law as its LSM.