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.

翻译：我们研究了属于高斯正交系综的 $m \times m$ 矩阵 $A_1,\ldots,A_n$ 的渐近差异，该系综是一类具有独立正态分布元素的随机对称矩阵。在 $m^2 = o(n)$ 的条件下，我们的结果表明，存在一个符号向量 $x \in \{\pm1\}^n$，使得 $\sum_{i=1}^n x_iA_i$ 的谱范数以高概率为 $\Theta(\sqrt{nm}4^{-(1 + o(1))n/m^2})$。这一结果是最优的，并证实了 Kunisky 和 Zhang 最近提出的猜想。