  The existence of supermassive black holes at high redshifts ($z\sim7$) is
difficult to accommodate in standard astrophysical scenarios. It has been shown
that dark matter models with a subdominant self-interacting component are able
to produce early seeds for supermassive black holes through the gravothermal
catastrophe. Previous studies used a fluid equation approach, requiring some
limiting assumptions. Here we reconsider the problem using $N$-body
gravitational simulations starting from the formation of the initial dark
matter halo. We consider both elastic and dissipative scattering, and elucidate
the interplay between the dark matter microphysics and subsequent accretion of
the black hole needed to match the properties of observed high redshift
supermassive black holes. We find a region of parameter space in which a small
component of self-interacting dark matter can produce the observed high
redshift supermassive black holes.
