  Calculations of excited states in Green's function formalism often invoke the
diagonal approximation, in which the quasiparticle states are taken from a
mean-field calculation. Here, we extend the stochastic approaches applied in
the many-body perturbation theory and overcome this limitation for large
systems in which we are interested in a small subset of states. We separate the
problem into a core subspace, whose coupling to the remainder of the system
environment is stochastically sampled. This method is exemplified on computing
hole injection energies into CO$_2$ on an extended gold surface with nearly
3000 electrons. We find that in the extended system, the size of the problem
can be compressed up to $95\%$ using stochastic sampling. This result provides
a way forward for self-consistent stochastic methods and determining Dyson
orbitals in large systems.
