  This article proposes a first analysis of kernel spectral clustering methods
in the regime where the dimension $p$ of the data vectors to be clustered and
their number $n$ grow large at the same rate. We demonstrate, under a $k$-class
Gaussian mixture model, that the normalized Laplacian matrix associated with
the kernel matrix asymptotically behaves similar to a so-called spiked random
matrix. Some of the isolated eigenvalue-eigenvector pairs in this model are
shown to carry the clustering information upon a separability condition
classical in spiked matrix models. We evaluate precisely the position of these
eigenvalues and the content of the eigenvectors, which unveil important
(sometimes quite disruptive) aspects of kernel spectral clustering both from a
theoretical and practical standpoints. Our results are then compared to the
actual clustering performance of images from the MNIST database, thereby
revealing an important match between theory and practice.
