The Host Galaxy of the Recoiling Black Hole Candidate in 3C 186: An Old Major Merger Remnant at the Center of a z = 1 Cluster

3C 186, a radio-loud quasar at z = 1.0685, was previously reported to have both velocity and spatial offsets from its host galaxy, and has been considered as a promising candidate for a gravitational wave recoiling black hole triggered by a black hole merger. Another possible scenario is that 3C 186 is in an ongoing galaxy merger, exhibiting a temporary displacement. In this study, we present analyses of new deep images from the Hubble Space Telescope WFC3-IR and Advanced Camera for Surveys, aiming to characterize the host galaxy and test this alternative scenario. We carefully measure the light-weighted center of the host and reveal a significant spatial offset from the quasar core (11.1 ± 0.1 kpc). The direction of the confirmed offset aligns almost perpendicularly to the radio jet. We do not find evidence of a recent merger, such as a young starburst in disturbed outskirts, but only marginal light concentration in F160W at ~30 kpc. The host consists of mature (≳200 Myr) stellar populations and one compact star-forming region. We compare with hydrodynamical simulations and find that those observed features are consistently seen in late-stage merger remnants. Taken together, those pieces of evidence indicate that the system is not an ongoing/young merger remnant, suggesting that the recoiling black hole scenario is still a plausible explanation for the puzzling nature of 3C 186.