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We discuss the latest results of numerical simulations following the orbitaldecay of massive black hole pairs in galaxy mergers. We highlight importantdifferences between gas-poor and gas-rich hosts, and between orbital evolutiontaking place at high redshift as opposed to low redshift. Two effects have ahuge impact and are rather novel in the context of massive black hole binaries.The first is the increase in characteristic density of galactic nuclei ofmerger remnants as galaxies are more compact at high redshift due to the waydark halo collapse depends on redshift. This leads naturally to hardeningtimescales due to 3-body encounters that should decrease by two orders ofmagnitude up to $z=4$. It explains naturally the short binary coalescencetimescale, $\sim 10$ Myr, found in novel cosmological simulations that followbinary evolution from galactic to milliparsec scales. The second one is theinhomogeneity of the interstellar medium in massive gas-rich disks at highredshift. In the latter star forming clumps 1-2 orders of magnitude moremassive than local Giant Molecular Clouds (GMCs) can scatter massive blackholes out of the disk plane via gravitational perturbations and directencounters. This renders the character of orbital decay inherently stochastic,often increasing orbital decay timescales by as much as a Gyr. At low redshifta similar regime is present at scales of $1-10$ pc inside Circumnuclear GasDisks (CNDs). In CNDs only massive black holes with masses below $10^7M_{\odot}$ can be significantly perturbed. They decay to sub-pc separations inup to $\sim 10^8$ yr rather than the in just a few million years as in a smoothCND. Finally implications for building robust forecasts of LISA event rates arediscussed

Multiple regimes and coalescence timescales for massive black hole pairs ; the critical role of galaxy formation physics