  Let (M, g) be a compact smooth Riemannian manifold. We obtain new
off-diagonal estimates as {\lambda} tend to infinity for the remainder in the
pointwise Weyl Law for the kernel of the spectral projector of the Laplacian
onto functions with frequency at most {\lambda}. A corollary is that, when
rescaled around a non self-focal point, the kernel of the spectral projector
onto the frequency interval (\lambda, \lambda + 1] has a universal scaling
limit as {\lambda} goes to infinity (depending only on the dimension of M). Our
results also imply that if M has no conjugate points, then immersions of M into
Euclidean space by an orthonormal basis of eigenfunctions with frequencies in
(\lambda, \lambda + 1] are embeddings for all {\lambda} sufficiently large.
