论文标题
GOE/GUE特征向量的二次形式的极端统计
Extremal statistics of quadratic forms of GOE/GUE eigenvectors
论文作者
论文摘要
我们考虑在大型$ n \ times n $ goe或gue矩阵的随机特征向量上评估的确定性矩阵$ a $的二次形式，或在HAAR-ORTORTOR-ORTORTONAL或HAAR-HAAR-INTIRATION随机随机矩阵的列进行评估。我们证明，只要确定性矩阵的排名远小于$ \ sqrt {n} $，这些二次形式的极端分布渐近地与特征向量是独立的高斯人相同。这将问题降低到高斯计算中，我们在几种情况下进行了以说明我们的结果，根据$ a $的签名，找到Gumbel或Weibull限制分布。我们的结果自然也适用于任何不变合奏的特征向量。
We consider quadratic forms of deterministic matrices $A$ evaluated at the random eigenvectors of a large $N \times N$ GOE or GUE matrix, or equivalently evaluated at the columns of a Haar-orthogonal or Haar-unitary random matrix. We prove that, as long as the deterministic matrix has rank much smaller than $\sqrt{N}$, the distributions of the extrema of these quadratic forms are asymptotically the same as if the eigenvectors were independent Gaussians. This reduces the problem to Gaussian computations, which we carry out in several cases to illustrate our result, finding Gumbel or Weibull limiting distributions depending on the signature of $A$. Our result also naturally applies to the eigenvectors of any invariant ensemble.
