Tracing black hole and galaxy growth across environments since cosmic noon

The distribution of systems in the black hole mass-stellar mass ($M_\mathrm{BH}-M_\star$) plane encodes not only the integrated growth of galaxies and their central black holes (BHs), but also the processes that shape their evolution. Using the ASTRID and TNG300 cosmological simulations, we track massive BHs from cosmic noon ($z=2$) to $z=0.5$, spanning 5.3 Gyr of assembly. Unlike most previous studies, we follow the BHs themselves rather than their original host galaxies, thereby capturing central BHs, BHs in satellites, and off-nuclear wandering BHs. We find that central BHs in both simulations evolve along a relatively tight, nearly redshift-invariant $M_\mathrm{BH}-M_\star$ relation that is broadly consistent with local empirical constraints and with measurements from variable active galactic nuclei (AGN) at comparable redshifts. Departures from this relation trace distinct evolutionary channels. High-mass central BHs grow substantially through mergers and subsequently quench their hosts through AGN kinetic feedback. Tidal stripping moves satellites to lower $M_\star$ at nearly fixed $M_\mathrm{BH}$, producing weakly accreting, overmassive central BHs in gas-poor systems. In ASTRID, satellite accretion and inefficient dynamical friction generate wandering BHs that are undermassive relative to their new hosts and experience minimal accretion or merger-driven growth. These populations occupy characteristic regions in both the $M_\mathrm{BH}-M_\star$ and the specific BH accretion rate-specific star formation rate planes, demonstrating that a BH's location in these planes is a fossil record of its dynamical, accretion, and feedback history.