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 $din(1/3,d_{mathrm{crit}})$ 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 $pin (1,infty)$ 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.
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