  We present an analysis of the instantaneous supermassive black hole (SMBH)
accretion rates in a collection of 1563 post-merger galaxies drawn from the
IllustrisTNG simulation. Our sample consists of galaxies that have experienced
a merger in the last simulation snapshot (within ~160 Myrs of coalescence) in
the redshift range 0<z<1, with merger stellar mass ratios >1:10 and post-merger
stellar masses > $10^{10} M_{\odot}$. We find that, on average, the accretion
rates of the post-mergers are ~1.7 times higher than in a control sample and
that post-mergers are 3-4 times more likely to experience a luminous active
galactic nuclei (AGN) phase than isolated galaxies. SMBH accretion rate
enhancements persist for ~2 Gyrs after coalescence, significantly exceeding the
~500 Myr lifetime of star formation rate enhancements. We find that the
presence of simultaneous enhancements in both the star formation and SMBH
accretion rates depends on both the mass ratio of the merger and on the gas
mass of the post-merger galaxy. Despite these accretion rate enhancements, only
~35% of post-mergers experience a luminous AGN ($L_{bol}>10^{44}$ erg/s) within
500 Myrs after coalescence, and fewer than 10\% achieve a luminosity in excess
of $L_{bol}>10^{45}$ erg/s. Moreover, only ~10\% of the highest luminosity
($L_{bol}>10^{45}$ erg/s) AGN in the IllustrisTNG galaxy sample are recent
mergers. Our results are therefore consistent with a picture in which mergers
can (but don't always) trigger AGN activity, but where the majority of galaxies
hosting high luminosity AGN are not recent mergers.
