In this article, we explore the spectral properties of general random kernel matrices $[K(U_i,U_j)]_{1\leq i\neq j\leq n}$ from a Lipschitz kernel $K$ with $n$ independent random variables $U_1,U_2,\ldots, U_n$ distributed uniformly over $[0,1]$. In particular we identify a dichotomy in the extreme eigenvalue of the kernel matrix, where, if the kernel $K$ is degenerate, the largest eigenvalue of the kernel matrix (after proper normalization) converges weakly to a weighted sum of independent chi-squared random variables. In contrast, for non-degenerate kernels, it converges to a normal distribution extending and reinforcing earlier results from Koltchinskii and Gin\'e (2000). Further, we apply this result to show a dichotomy in the asymptotic behavior of extreme eigenvalues of $W$-random graphs, which are pivotal in modeling complex networks and analyzing large-scale graph behavior. These graphs are generated using a kernel $W$, termed as graphon, by connecting vertices $i$ and $j$ with probability $W(U_i, U_j)$. Our results show that for a Lipschitz graphon $W$, if the degree function is constant, the fluctuation of the largest eigenvalue (after proper normalization) converges to the weighted sum of independent chi-squared random variables and an independent normal distribution. Otherwise, it converges to a normal distribution.

翻译：本文研究Lipschitz核$K$生成的一般随机核矩阵$[K(U_i,U_j)]_{1\leq i\neq j\leq n}$的谱性质，其中$U_1,U_2,\ldots,U_n$为$[0,1]$上均匀分布的独立随机变量。我们揭示了核矩阵极端特征值的一个二分现象：若核$K$是退化的，则核矩阵最大特征值（经适当标准化后）弱收敛至独立卡方随机变量的加权和；反之，对于非退化核，其收敛于正态分布，这一结果推广并强化了Koltchinskii与Giné (2000)的早期结论。进一步，我们将此结果应用于$W$-随机图，证明其极端特征值的渐近行为同样呈现二分现象。这类图通过核函数$W$（即图朗）生成：顶点$i$与$j$以概率$W(U_i,U_j)$相连，在复杂网络建模与大规模图行为分析中具有关键作用。我们的结果表明：对于Lipschitz图朗$W$，若度函数为常数，则最大特征值的波动（经适当标准化后）收敛于独立卡方随机变量的加权和与一个独立正态分布的叠加；反之则收敛于正态分布。