论文标题
关于不均匀随机矩阵ESD的通用性的注释
A note on the universality of ESDs of inhomogeneous random matrices
论文作者
论文摘要
简而言之，我们通过表明技术和难以验证的傅立叶统治性可以简单地被天然统一的抗聚会假设替换，将Tao和Vu的著名结果以及Krishnapur关于经验光谱分布的普遍性扩展到了广泛的不均匀的复杂随机矩阵。 在此过程中，我们表明，不均匀的复杂随机矩阵，其预期的平方Hilbert-schmidt Norm在维度上是二次的，其内脏（在对称后）均匀地抗凝结为$ 0 $，而无限度则是$ 0 $ and Infinity，通常具有最小的奇异值$ω（n^{-1/2}）$。费率$ n^{ - 1/2} $很清晰，并在文献中缩小了差距。 我们的证据紧密遵循Livshyts的最新作品，Livshyts，Tikhomirov和Vershynin在不均匀的真实随机矩阵上。新成分是几个抗浓缩不平等的不等式，但不一定是独立的，复杂的随机变量，这在其他情况下也可能有用。
In this short note, we extend the celebrated results of Tao and Vu, and Krishnapur on the universality of empirical spectral distributions to a wide class of inhomogeneous complex random matrices, by showing that a technical and hard-to-verify Fourier domination assumption may be replaced simply by a natural uniform anti-concentration assumption. Along the way, we show that inhomogeneous complex random matrices, whose expected squared Hilbert-Schmidt norm is quadratic in the dimension, and whose entries (after symmetrization) are uniformly anti-concentrated at $0$ and infinity, typically have smallest singular value $Ω(n^{-1/2})$. The rate $n^{-1/2}$ is sharp, and closes a gap in the literature. Our proofs closely follow recent works of Livshyts, and Livshyts, Tikhomirov, and Vershynin on inhomogeneous real random matrices. The new ingredient is a couple of anti-concentration inequalities for sums of independent, but not necessarily identically distributed, complex random variables, which may also be useful in other contexts.
