Asymptotic Discrepancy of Gaussian Orthogonal Ensemble Matrices

Abstract: We study the asymptotic discrepancy of $m \times m$ matrices $A_1,\ldots,A_n$ belonging to the Gaussian orthogonal ensemble, which is a class of random symmetric matrices with independent normally distributed entries. In the setting $m^2 = o(n)$, our results show that there exists a signing $x \in {\pm1}^n$ such that the spectral norm of $\sum_{i=1}^n x_iA_i$ is $\Theta(\sqrt{nm}4^{-(1 + o(1))n/m^2})$ with high probability. This is best possible and settles a recent conjecture by Kunisky and Zhang.