  We introduce a model for random groups in varieties of $n$-periodic groups as
$n$-periodic quotients of triangular random groups. We show that for an
explicit $d_{\mathrm{crit}}\in(1/3,1/2)$, for densities
$d\in(1/3,d_{\mathrm{crit}})$ and for $n$ large enough, the model produces
\emph{infinite} $n$-periodic groups. As an application, we obtain, for every
fixed large enough $n$, for every $p\in (1,\infty)$ an infinite $n$-periodic
group with fixed points for all isometric actions on $L^p$-spaces. Our main
contribution is to show that certain random triangular groups are uniformly
acylindrically hyperbolic.
