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.

考虑一个随机方阵，它具有均值零和单位方差的独立同分布条目。我们表明，随着维度趋于无穷大，谱半径在概率上等于维度的平方根。这个结果也可以看作是循环定律定理中支点在最优矩条件下的收敛。在证明中，我们建立了倒数特征多项式对单位圆盘外随机解析函数的收敛性，与双曲高斯解析函数相关。证明很简短，与光谱半径的通常方法不同。它依赖于紧性论证和联合中心极限现象来确定固定幂的痕迹。