论文标题
随机基质的光谱半径通过其特征多项式的收敛性
Convergence of the spectral radius of a random matrix through its characteristic polynomial
论文作者
论文摘要
考虑一个平均零和单位方差的独立和相同分布的条目的正方形随机矩阵。我们表明，随着尺寸倾向于无穷大，光谱半径等于概率的尺寸的平方根。该结果也可以看作是在最佳时刻条件下循环定理中支持的收敛。在证明中，我们建立了与单位盘外部的随机分析函数的相互特征多项式法律的收敛性，这与双曲线高斯分析函数有关。证明是简短的，与光谱半径的通常方法不同。它依靠一个紧密的论点和固定力痕迹的关节中心极限现象。
Consider a square random matrix with independent and identically distributed entries of mean zero and unit variance. We show that as the dimension tends to infinity, the spectral radius is equivalent to the square root of the dimension in probability. This result can also be seen as the convergence of the support in the circular law theorem under optimal moment conditions. In the proof we establish the convergence in law of the reciprocal characteristic polynomial to a random analytic function outside the unit disc, related to a hyperbolic Gaussian analytic function. The proof is short and differs from the usual approaches for the spectral radius. It relies on a tightness argument and a joint central limit phenomenon for traces of fixed powers.
