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The origin of the Binary Black Hole (BBH) mergers detected throughGravitational Waves (GWs) by the LIGO-Virgo-KAGRA (LVK) collaboration remainsdebated. One fundamental reason is our ignorance of their host environment, asthe typical size of an event's localization volume can easily contain thousandsof galaxies. A strategy around this is to exploit statistical approaches toassess the spatial correlation between these mergers and astrophysicallymotivated host galaxy types, such as Active Galactic Nuclei (AGN). We use aLikelihood ratio method to infer the degree of GW-AGN connection out to$z=0.2$. We simulate BBH mergers whose components' masses are sampled from arealistic distribution of the underlying population of Black Holes (BHs).Localization volumes for these events are calculated assuming two differentinterferometric network configurations. These correspond to the configurationof the third (O3) and of the upcoming fourth (O4) LVK observing runs. Weconclude that the 13 BBH mergers detected during the third observing run at$z\leq0.2$ are not enough to reject with a $3\sigma$ significance thehypothesis according to which there is no connection between GW and AGN moreluminous than $\approx 10^{44.3}\rm{erg}\ \rm{s}^{-1}$, that have numberdensity higher than $10^{-4.75}\textrm{Mpc}^{-3}$. However, 13 detections areenough to reject this no-connection hypothesis when rarer categories of AGN areconsidered, with bolometric luminosities greater than $\approx10^{45.5}\rm{erg}\ \rm{s}^{-1}$. We estimate that O4 results will potentiallyallow us to test fractional contributions to the total BBH merger populationfrom AGN of any luminosity higher than $80\%$.

Detectability of a spatial correlation between stellar-mass black hole mergers and Active Galactic Nuclei in the Local Universe