  We report on a study of a spin-down impurity strongly coupled to a spin-up
Fermi sea (a so-called Fermi polaron) with the diagrammatic Monte-Carlo
(DiagMC) technique. Conditions of zero temperature and three dimensions are
considered for an ultracold atomic gas with resonant interactions in the
zero-range limit. A Feynman diagrammatic series is developed for the one-body
and two-body propagators providing information about the polaron and molecule
channel respectively. The DiagMC technique allows us to reach diagram orders
that are high enough for extrapolation to infinite order. The robustness of the
extracted results is examined by checking various resummation techniques and by
running the simulations with various choices for the propagators and vertex
functions. It turns out that dressing the lines in the diagrams as much as
possible is not always the optimal choice. We also identify classes of dominant
diagrams for the one-body and two-body self-energy in the region of strong
interaction. These dominant diagrams turn out to be the leading processes of
the strong-coupling limit. The quasiparticle energies and $Z$-factor are
obtained as a function of the interaction strength. We find that the DiagMC
results for the molecule and polaron properties are very similar to those
obtained with a variational ansatz. Surprisingly, this variational ansatz gives
very good predictions for the quasiparticle residue even when this residue is
significantly smaller than one.
