The early third data release (EDR3) of the European Space Agency satellite Gaia provides coordinates, parallaxes, and proper motions for ~1.47 billion sources in our Milky Way, based on 34 months of observations. The combination of Gaia DR2 radial velocities with the more precise and accurate astrometry provided by Gaia EDR3 makes the best dataset available to search for the fastest nearby stars in our Galaxy. We compute the velocity distribution of ~7 million stars with precise parallaxes, to investigate the high-velocity tail of the velocity distribution of stars in the Milky Way. We release a catalogue with distances, total velocities, and corresponding uncertainties for all the stars considered in our analysis, available at https://sites.google.com/view/tmarchetti/research . By applying quality cuts on the Gaia astrometry and radial velocities, we identify a clean subset of 94 stars with a probability Pub > 50% to be unbound from our Galaxy. 17 of these have Pub > 80% and are our best candidates. We propagate these stars in the Galactic potential to characterize their orbits. We find that 11 stars are consistent with being ejected from the Galactic disk, and are possible hyper-runaway star candidates. The other 6 stars are not consistent with coming from a known star-forming region. We investigate the effect of adopting a parallax zero point correction, which strongly impacts our results: when applying this correction, we identify only 12 stars with Pub > 50%, 3 of these having Pub > 80%. Spectroscopic follow-ups with ground-based telescopes are needed to confirm the candidates identified in this work.
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