Black Hole Discs and Spheres in Galactic Nuclei -- Exploring the
  Landscape of Vector Resonant Relaxation Equilibria
  Vector resonant relaxation (VRR) is known to be the fastest gravitational
process that shapes the geometry of stellar orbits in nuclear star clusters.
This leads to the realignment of the orbital planes on the corresponding VRR
time scale $t_{\rm VRR}$ of a few million years, while the eccentricity $e$ and
semimajor axis $a$ of the individual orbits are approximately conserved. The
distribution of orbital inclinations reaches an internal equilibrium
characterised by two conserved quantities, the total potential energy among
stellar orbits, $E_{\rm tot}$, and the total angular momentum, $L_{\rm tot}$.
On timescales longer than $t_\mathrm{VRR}$, the eccentricities and semimajor
axes change slowly and the distribution of orbital inclinations are expected to
evolve through a series of VRR equilibria. Using a Monte Carlo Markov Chain
method, we determine the equilibrium distribution of orbital inclinations in
the microcanonical ensemble with fixed $E_{\rm tot}$ and $L_{\rm tot}$ for
isolated nuclear star clusters with a power-law distribution of $a$, $e$, and
$m$, where $m$ is the stellar mass. We explore the possible equilibria for $9$
representative $E_{\rm tot}$--$L_{\rm tot}$ pairs that cover the possible
parameter space. For all cases, the equilibria show anisotropic mass
segregation where the distribution of more massive objects is more flattened
than that for lighter objects. Given that stellar black holes are more massive
than the average main sequence stars, these findings suggest that black holes
reside in disc-like structures within nuclear star clusters for a wide range of
initial conditions.
