  In this paper, we study the spectrum and the eigenvectors of radial kernels
for mixtures of distributions in $\mathbb{R}^n$. Our approach focuses on high
dimensions and relies solely on the concentration properties of the components
in the mixture. We give several results describing of the structure of kernel
matrices for a sample drawn from such a mixture. Based on these results, we
analyze the ability of kernel PCA to cluster high dimensional mixtures. In
particular, we exhibit a specific kernel leading to a simple spectral algorithm
for clustering mixtures with possibly common means but different covariance
matrices. We show that the minimum angular separation between the covariance
matrices that is required for the algorithm to succeed tends to $0$ as $n$ goes
to infinity.
