  Understanding the formation and growth of supermassive black holes (SMBHs) at
high redshift represents a major challenge for theoretical models. In this work
we investigate the early evolution of the first SMBHs by constraining their
distribution in mass and luminosity at $z > 4$. In particular, we focus on the
poorly explored low-mass end of the nuclear black hole (BH) distribution down
to $z \simeq 4$, and explore its connection with the nature of the first BH
seeds and the processes governing their mass growth. To this aim, we have
developed CAT (Cosmic Archaeology Tool), a new semi-analytic model that
describes the formation of the first stars and black holes in a self-consistent
way and follows the co-evolution of nuclear BHs and their host galaxies for a
representative population at $z > 4$. We find that current observational
constraints favour models where the growth of BH seeds is Eddington limited and
occurs at the Bondi-Hoyle-Lyttleton rate or where super-Eddington accretion
occurs via a slim disk during gas rich galaxy mergers. The main difference
between these two model variants lies at the low-end of the predicted mass and
luminosity functions at $4 \le z \le 6$, where a clear gap appears in the first
model, reflecting the stunted growth of light BH seeds formed as remnants of
the first stars. Detecting this signature will be extremely challenging even
for the future generation of space observatories, such as JWST, Athena and
Lynx.
