AI 中文总结
本文研究尾部指数α∈[0,2]的自归一化随机矩阵特征多项式，证明其在单位圆盘外收敛于泊松点过程描述的乘性混沌F_α，存在泊松与高斯混沌的普适 regime 转变，并推导得谱半径依概率渐近上界为1。
AI 中文摘要
我们研究自归一化随机矩阵的特征多项式，这类矩阵的行相互独立且归一化后具有单位L²范数。归一化前的元素被假定具有正则变化尾部，尾部指数α∈[0,2]。我们证明，在单位圆盘外，该特征多项式收敛于随机解析函数F_α。我们将F_α识别为由泊松点过程描述的乘性混沌。极限函数族(F_α)_{α∈[0,2]}在两种普适 regime 之间插值：α=0时为泊松乘性混沌，边界α=2时为高斯乘性混沌。因此，自归一化提供了一种矩阵模型，其中随着尾部指数变化，可观察到极限特征多项式在泊松 regime 与高斯 regime 之间的转变。在自归一化矩阵的迹的波动中也发现了类似转变。作为我们结果的一项应用，我们推导得出：对于任何对称元素分布，自归一化矩阵的谱半径依概率渐近上界为1。
英文摘要
We study the characteristic polynomial of self-normalized random matrices, whose rows are independent and normalized to have unit $\mathrm{L}^2$ norm. The entries before normalization are assumed to have regularly varying tails with tail index $α\in [0,2]$. We prove that, outside the unit disk, the characteristic polynomial converges to a random analytic function $F_α$. We identify $F_α$ as a multiplicative chaos described in terms of Poisson point processes. The family of limiting functions $(F_α)_{α\in[0,2]}$ interpolates between two universal regimes: Poisson multiplicative chaos at $α=0$ and Gaussian multiplicative chaos at the boundary $α=2$. Thus, self-normalization provides a matrix model where one observes the transition between Poissonian and Gaussian regimes for the limiting characteristic polynomial as the tail index varies. A similar transition is found for the fluctuations of the traces of self-normalized matrices. As an application of our results, we derive that the spectral radius of self-normalized matrices is asymptotically bounded above by one in probability for any symmetric entry distribution.
