  The Hartree-Fock approximation of Quantum Electrodynamics provides a rigorous
framework for the description of relativistic electrons in external fields.
This nonlinear model takes into account the infinitely many virtual electrons
of Dirac's vacuum as well as the Coulomb interactions between all the
particles. The state of the system is an infinite-rank projection satisfying a
nonlinear equation. In this paper, we construct solutions to this equation, in
the regime of weak interactions (that is, small coupling constant $\alpha$),
and strong external fields (that is, large atomic charge $Z$ such that $\alpha
Z:=\kappa$ stays fixed). In this regime, we are able to remove the ultraviolet
cut-off $\Lambda$ as soon as $\alpha\log\Lambda$ stays fixed. As an application
of this result, we compare the critical strength $\kappa_c(\alpha)$ of the
external potential needed to produce an additional particle in the vacuum, when
$\alpha=0$ or $\alpha>0$. We prove that
$\lim_{\alpha\to0}\kappa_c(\alpha)/\kappa_c(0)> 1$, and we identify the limit
exactly. Because of the dielectric behavior of Dirac's vacuum, static
electron/positron pair production occurs in the interacting case for a stronger
field that in the non-interacting case, which is a mere consequence of charge
renormalization.
