论文标题
随机统一矩阵痕迹的多元正常近似
Multivariate normal approximation for traces of random unitary matrices
论文作者
论文摘要
在本文中，我们在第一个$ m $ powers the the $ n \ times n $ n $随机统一矩阵和200万美元$ $ $维的高斯随机变量之间的痕迹之间达到了超数的收敛速率。这将标量案例中的先前结果概括为多元设置，我们还以明确的常数为估算中的尺寸$ m $和$ n $的确切依赖性。我们对$ n $增长的制度特别感兴趣，我们的主要结果基本上指出，如果$ m \ ll \ sqrt {n} $，那么高斯近似值的收敛速率为$γ（\ frac nm+1）我们还表明，高斯近似对于所有$ m \ ll n^{2/3} $仍然有效，没有快速的收敛速度。
In this article, we obtain a super-exponential rate of convergence in total variation between the traces of the first $m$ powers of an $n\times n$ random unitary matrices and a $2m$-dimensional Gaussian random variable. This generalizes previous results in the scalar case to the multivariate setting, and we also give the precise dependence on the dimensions $m$ and $n$ in the estimate with explicit constants. We are especially interested in the regime where $m$ grows with $n$ and our main result basically states that if $m\ll \sqrt{n}$, then the rate of convergence in the Gaussian approximation is $Γ(\frac nm+1)^{-1}$ times a correction. We also show that the Gaussian approximation remains valid for all $m\ll n^{2/3}$ without a fast rate of convergence.
