LOCAL INHOMOGENEOUS CIRCULAR LAW
LOCAL INHOMOGENEOUS CIRCULAR LAW
复制标题
DOI：
10.1214/17-aap1302
复制
发表时间：
2018-02-01
影响因子：
1.8
通讯作者：
Krueger, Torben
中科院分区：
文献类型：
--
作者：
Alt, Johannes;Erdos, Laszlo;Krueger, Torben
我们考虑具有中心独立元素的大型随机矩阵X，它们具有可比但不一定相同的方差。Girko的循环定律断言，光谱在圆盘中是支持的，并且在相同方差的情况下，极限密度是均匀的。在这种特殊的情况下，当地循环法由Bournovel等人[Probab. Theory Related Fields 159（2014）545-595; Probab. Theory Related Fields 159（2014）619-660]表明，经验密度甚至在略高于典型特征值间距的尺度上局部收敛。在一般情况下，极限密度通常是不均匀的，它是通过求解确定性方程组获得的。我们的主要结果是局部非齐次循环律在最佳尺度上的整体谱的一般方差轮廓的项目X。
We consider large random matrices X with centered, independent entries, which have comparable but not necessarily identical variances. Girko's circular law asserts that the spectrum is supported in a disk and in case of identical variances, the limiting density is uniform. In this special case, the local circular law by Bourgade et al. [Probab. Theory Related Fields 159 (2014) 545-595; Probab. Theory Related Fields 159 (2014) 619-660] shows that the empirical density converges even locally on scales slightly above the typical eigenvalue spacing. In the general case, the limiting density is typically inhomogeneous and it is obtained via solving a system of deterministic equations. Our main result is the local inhomogeneous circular law in the bulk spectrum on the optimal scale for a general variance profile of the entries of X.