  We introduce Gaussian-type measures on the manifold of all metrics with a
fixed volume form on a compact Riemannian manifold of dimension $\geq 3$. For
this random model we compute the characteristic function for the $L^2$ (Ebin)
distance to the reference metric. In the Appendix, we study Lipschitz-type
distance between Riemannian metrics and give applications to the diameter,
eigenvalue and volume entropy functionals.
