  The Hartree-Fock equation admits homogeneous states that model infinitely
many particles at equilibrium. We prove their asymptotic stability in large
dimensions, under assumptions on the linearised operator. Perturbations are
moreover showed to scatter to linear waves. We obtain this result for the
equivalent formulation of the Hartree-Fock equation in the framework of random
fields. The main novelty is to study the full Hartree-Fock equation, including
for the first time the exchange term in the study of these stationary
solutions.
