  Gromov showed that (1993) with high probability, every bounded and reduced
van Kampen diagram $D$ of a random group at density $d$ satisfies the
isoperimetric inequality $|\partial D|\geq (1-2d-s)|D|\ell$. In this article,
we adapt Gruber-Mackay's prove for random triangular groups, showing a
non-reduced 2-complex version of this inequality.
  Moreover, for any 2-complex $Y$ of a given geometric form, we exhibit a phase
transition: we give explicitly a critical density $d_c$ depending only on $Y$
such that, in a random group at density $d$, if $d<d_c$ then there is no
reduced van Kampen 2-complex of the form $Y$; while if $d>d_c$ then there
exists reduced van Kampen 2-complexes of the form $Y$.
  As an application, we show a phase transition for the $C(p)$
small-cancellation condition: for a random group at density $d$, if $d<1/(p+1)$
then it satisfies $C(p)$; while if $d>1/(p+1)$ then it does not satisfy $C(p)$.
