  Understanding the formation and evolution of giant planets ($\ge$1 $M_{Jup}$)
at wide orbital separation ($\ge$5 AU) is one of the goals of direct imaging.
Over the past 15 years, many surveys have placed strong constraints on the
occurrence rate of wide-orbit giants, mostly based on non-detections, but very
few have tried to make a direct link with planet formation theories. In the
present work, we combine the results of our previously published VLT/NaCo large
program with the results of 12 past imaging surveys to constitute a statistical
sample of 199 FGK stars within 100 pc, including 3 stars with sub-stellar
companions. Using Monte Carlo simulations and assuming linear flat
distributions for the mass and semi-major axis of planets, we estimate the
sub-stellar companion frequency to be within 0.75-5.7% at the 68% confidence
level (CL) within 20-300 AU and 0.5-75 $M_{Jup}$, which is compatible with
previously published results. We also compare our results with the predictions
of state-of-the-art population synthesis models based on the gravitational
instability (GI) formation scenario by Forgan & Rice (2013), with and without
scattering. We estimate that in both the scattered and non-scattered
populations, we would be able to detect more than 30% of companions in the 1-75
$M_{Jup}$ range (95% CL). With the 3 sub-stellar detections in our sample, we
estimate the fraction of stars that host a planetary system formed by GI to be
within 1.0-8.6% (95% CL). We also conclude that even though GI is not common,
it predicts a mass distribution of wide-orbit massive companions that is much
closer to what is observed than what the core accretion scenario predicts.
Finally, we associate the present paper with the release of the Direct Imaging
Virtual Archive (DIVA, http://cesam.lam.fr/diva/), a public database that aims
at gathering the results of past, present, and future direct imaging surveys.
