  We provide two constructions of hyperbolic metrics on 3-manifolds with
Heegaard splittings that satisfy certain topological conditions, which both
apply to random Heegaard splittings with asymptotic probability 1.
  These constructions provide a lot of control on the resulting metric,
allowing us to prove various results about the coarse growth rate of geometric
invariants, such as diameter and injectivity radius, and about arithmeticity
and commensurability in families of random 3-manifolds. For example, we show
that the diameter of a random Heegaard splitting grows coarsely linearly in the
length of the associated random walk.
  The constructions only use tools from the deformation theory of Kleinian
groups, that is, we do not rely on the solution of the Geometrization
Conjecture by Perelman. In particular, we give a proof of Maher's result that
random 3-manifolds are hyperbolic that bypasses Geometrization.
