  We analyze the spectral clustering procedure for identifying coarse structure
in a data set $x_1, \dots, x_n$, and in particular study the geometry of graph
Laplacian embeddings which form the basis for spectral clustering algorithms.
More precisely, we assume that the data is sampled from a mixture model
supported on a manifold $\mathcal{M}$ embedded in $\mathbb{R}^d$, and pick a
connectivity length-scale $\varepsilon>0$ to construct a kernelized graph
Laplacian. We introduce a notion of a well-separated mixture model which only
depends on the model itself, and prove that when the model is well separated,
with high probability the embedded data set concentrates on cones that are
centered around orthogonal vectors. Our results are meaningful in the regime
where $\varepsilon = \varepsilon(n)$ is allowed to decay to zero at a slow
enough rate as the number of data points grows. This rate depends on the
intrinsic dimension of the manifold on which the data is supported.
