论文标题
半重量尾部矩阵线性光谱统计的协方差内核
Covariance Kernel of Linear Spectral Statistics for Half-Heavy tailed Wigner Matrices
论文作者
论文摘要
在本文中，我们分析了高斯过程的协方差内核，该过程是Wigner矩阵的线性光谱统计波动的极限。更确切地说，我们在这里研究的过程对应于具有独立条目的遗传矩阵，其$α$矩的时间为$ 2 <α<4 $。我们通过明确整合文献中获得的已知双Laplace Transform Antegral公式，从而获得了由Stieltjes变换的波动而导致的限制过程的协方差的封闭形式。然后，我们将协方差表示为作用于有限连续测试函数的积分内核。最终的公式使我们能够对该基质合奏对协方差结构的典型大型特征值进行启发式解释。
In this paper we analyze the covariance kernel of the Gaussian process that arises as the limit of fluctuations of linear spectral statistics for Wigner matrices with a few moments. More precisely, the process we study here corresponds to Hermitian matrices with independent entries that have $α$ moments for $2<α< 4$. We obtain a closed form $α$-dependent expression for the covariance of the limiting process resulting from fluctuations of the Stieltjes transform by explicitly integrating the known double Laplace transform integral formula obtained in the literature. We then express the covariance as an integral kernel acting on bounded continuous test functions. The resulting formulation allows us to offer a heuristic interpretation of the impact the typical large eigenvalues of this matrix ensemble have on the covariance structure.
