随机矩阵中含少量独立元素的半圆律
Semicircular law with a few independent entries in a random matrix
AI总结：
本文针对随机矩阵领域的半圆律问题，打破Wigner矩阵需O(n²)个独立随机变量的固有认知，构造出仅含O(n)个独立随机变量、经验谱分布可任意接近半圆律的矩阵。
AI中文摘要：
在随机矩阵领域，众所周知Wigner矩阵的极限谱分布为半圆律，而Toeplitz、Hankel、对称循环矩阵及逆循环矩阵等其他带模式矩阵的极限谱分布支撑集无界。Wigner矩阵与上述其他矩阵的一个根本区别是：Wigner矩阵有O(n²)个独立随机变量，而其他矩阵仅有O(n)个独立随机变量。本文证明该结论不具一般性，特别地，我们构造了具有O(n)个独立随机变量的矩阵，其经验谱分布可任意接近半圆律。
英文摘要：
It is well known in random matrix literature that the limiting spectral distribution of a Wigner matrix is the semi circular law while the limiting spectral distributions of other patterned matrices like Toeplitz, Hankel, symmetric circulant and reverse circulant matrices have unbounded supports. One fundamental difference between the Wigner matrices and the other matrices mentioned above is that the Wigner matrices have $O(n^2)$ independent random variables while the others have $O(n)$ independent random variables. In this paper, we show that this is not true in general. In particular, we form matrices with $O(n)$ independent random variables whose empirical spectral distributions are arbitrary close to the semi circular law.
