Growth exponent of generic groups

In [GrH97], Grigorchuk and de la Harpe ask for conditions under which some group presentations have growth rate close to that of the free group with the same number of generators. We prove that this property holds for a generic group (in the density model of random groups). Namely, for every positive E, the property of having growth exponent at least 1 - epsilon (in base 2m - I where m is the number of generators) is generic in this model. In particular this extends a theorem of Shukhov [Shu99]. More generally, we prove that the growth exponent does not change much through a random quotient of a torsion-free hyperbolic group.