  The Hartree-Fock ground state of the Homogeneous Electron Gas is never
translation invariant, even at high densities. As proved by Overhauser, the
(paramagnetic) free Fermi Gas is always unstable under the formation of spin or
charge density waves. We give here the first explicit bound on the energy gain
due to the breaking of translational symmetry. Our bound is exponentially small
at high density, which justifies posteriori the use of the non-interacting
Fermi Gas as a reference state in the large-density expansion of the
correlation energy of the Homogeneous Electron Gas. We are also able to discuss
the positive temperature phase diagram and prove that the Overhauser
instability only occurs at temperatures which are exponentially small at high
density. Our work sheds a new light on the Hartree-Fock phase diagram of the
Homogeneous Electron Gas.
