  In the density model of random groups, we consider presentations with any
fixed number m of generators and many random relators of length l, sending l to
infinity. If d is a "density" parameter measuring the rate of exponential
growth of the number of relators compared to the length of relators, then many
group-theoretic properties become generically true or generically false at
different values of d. The signature theorem for this density model is a phase
transition from triviality to hyperbolicity: for d < 1/2, random groups are
a.a.s. infinite hyperbolic, while for d > 1/2, random groups are a.a.s. order
one or two. We study random groups at the density threshold d = 1/2. Kozma had
found that trivial groups are generic for a range of growth rates at d = 1/2;
we show that infinite hyperbolic groups are generic in a different range. (We
include an exposition of Kozma's previously unpublished argument, with slightly
improved results, for completeness.)
