We consider certain large random matrices, called random inner-product kernel matrices, which are essentially given by a nonlinear function $f$ applied entrywise to a sample-covariance matrix, $f(X^TX)$, where $X \in \mathbb{R}^{d \times N}$ is random and normalized in such a way that $f$ typically has order-one arguments. We work in the polynomial regime, where $N \asymp d^\ell$ for some $\ell > 0$, not just the linear regime where $\ell = 1$. Earlier work by various authors showed that, when the columns of $X$ are either uniform on the sphere or standard Gaussian vectors, and when $\ell$ is an integer (the linear regime $\ell = 1$ is particularly well-studied), the bulk eigenvalues of such matrices behave in a simple way: They are asymptotically given by the free convolution of the semicircular and Mar\v{c}enko-Pastur distributions, with relative weights given by expanding $f$ in the Hermite basis. In this paper, we show that this phenomenon is universal, holding as soon as $X$ has i.i.d. entries with all finite moments. In the case of non-integer $\ell$, the Mar\v{c}enko-Pastur term disappears (its weight in the free convolution vanishes), and the spectrum is just semicircular.

翻译：本文研究一类称为随机内积核矩阵的大型随机矩阵，这类矩阵本质上是将非线性函数$f$逐元素应用于样本协方差矩阵$f(X^TX)$得到的，其中$X \in \mathbb{R}^{d \times N}$是随机矩阵且经过归一化使得$f$的自变量通常为阶-1量。我们工作在多项式机制下，即$N \asymp d^\ell$，其中$\ell > 0$，而不仅限于$\ell=1$的线性机制。先前多位学者的工作表明，当$X$的列向量服从球面均匀分布或标准高斯向量分布，且$\ell$为整数时（线性机制$\ell=1$已被充分研究），这类矩阵的谱主体特征呈现简单形式：其渐近谱由半圆分布与马尔琴科-帕斯图尔分布的自由卷积给出，权重系数由$f$的埃尔米特基展开确定。本文证明该现象具有普适性，只要$X$具有独立同分布元素且各阶矩有限即可成立。对于非整数$\ell$的情形，马尔琴科-帕斯图尔项消失（其在自由卷积中的权重归零），谱分布退化为纯半圆律。