Inference of recoil kicks from binary black hole mergers up to GWTC--4 and their astrophysical implications

We infer recoil (kick) velocities for all binary black hole merger events reported up to the GWTC–4 catalog, together with candidate intermediate-mass black hole events. We obtain informative kick constraints for GW231028_153006 ($839_{-681}^{+1018}\mathrm{k}\mathrm{m}{\mathrm{s}}^{-1}$) and GW231123_135430 ($974_{-760}^{+944}\mathrm{k}\mathrm{m}{\mathrm{s}}^{-1}$). Additionally, we compute recoil velocities for recently reported events from the ongoing fourth observing run: GW241011_233834, GW241110_124123, and GW250114_082203, obtaining $v_{\mathrm{k}\mathrm{i}\mathrm{c}\mathrm{k}}=974_{-466}^{+555}\mathrm{k}\mathrm{m}{\mathrm{s}}^{-1}$, $394_{-207}^{+582}\mathrm{k}\mathrm{m}{\mathrm{s}}^{-1}$, and $115_{-95}^{+301}\mathrm{k}\mathrm{m}{\mathrm{s}}^{-1}$, respectively. The remnant of GW241011_233834 is therefore inferred to have one of the largest recoil velocities among currently known events. We find that present recoil kick constraints are driven primarily by measurements of the mass ratio and spin magnitudes, while the contribution from spin orientation angles remains subdominant in most cases. We estimate typical retention probabilities of the remnant black holes in GWTC catalogs to be $\sim 1$–$5\mathrm{\%}$ for globular clusters, $\sim 15$–$30\mathrm{\%}$ for nuclear star clusters, $\sim 5$–$40\mathrm{\%}$ for dwarf galaxies, and $\sim 70$–$100\mathrm{\%}$ for elliptical galaxies. We further show that, even for remnants retained in globular clusters, recoil-induced spatial displacements from the cluster core are often significant, which can substantially suppress the chances of hierarchical mergers. We find that the probability for a GWTC merger remnant to participate in hierarchical mergers is $\sim 0.1$–$1\mathrm{\%}$ in globular clusters and $\sim 1$–$15\mathrm{\%}$ in nuclear star clusters.