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：本文探讨了利普希茨核矩阵$[K(U_i,U_j)]_{1\leq i\neq j\leq n}$的谱性质，其中核函数$K$为利普希茨连续，$U_1,U_2,\ldots, U_n$为独立同分布于$[0,1]$均匀分布的随机变量。我们特别揭示了该核矩阵极端特征值的二分现象：当核函数$K$退化时，经适当标准化后的核矩阵最大特征值弱收敛于独立卡方随机变量的加权和；而对于非退化核函数，其最大特征值则依分布收敛于正态分布，这一结果扩展并强化了Koltchinskii与Giné（2000）的早期结论。进一步地，我们将该结果应用于$W$-随机图极端特征值的渐近行为研究——这类图通过图论核$W$生成，以概率$W(U_i, U_j)$连接顶点$i$与$j$，对于复杂网络建模和大规模图行为分析具有关键作用。研究表明：对于利普希茨图论核$W$，若度函数为常数，则经适当标准化后的最大特征值波动收敛于独立卡方随机变量的加权和与独立正态分布的组合；否则收敛于正态分布。