Random triangular Burnside groups

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 $d_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}\in (1\mathrm{/}3,1\mathrm{/}2)$, for densities $d\in (1\mathrm{/}3,d_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}})$ 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 $p\in (1,\mathrm{\infty })$ 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.