  Consider the ensemble of Gaussian random potentials $\{V^L(q)\}_{L=1}^\infty$
on the $d$-dimensional torus where, essentially, $V^L(q)$ is a real-valued
trigonometric polynomial of degree $L$ whose coefficients are independent
standard normal variables. Our main result ensures that, with a probability
tending to 1 as $L\to\infty$, the dynamical system associated with the natural
Hamiltonian function defined by this random potential, $H^L:=\frac12|p|^2+
V^L(q)$, exhibits a number of chaotic regions which coexist with a
positive-volume set of invariant tori. In particular, these systems are
typically neither integrable with non-degenerate first integrals nor ergodic.
An analogous result for random natural Hamiltonian systems defined on the
cotangent bundle of an arbitrary compact Riemannian manifold is presented too.
