On The Random Turán number of linear cycles

Given two $r$-uniform hypergraphs $G$ and $H$ the Turán number $\mathrm{e}\mathrm{x}(G,H)$ is the maximum number of edges in an $H$-free subgraph of $G$. We study the typical value of $\mathrm{e}\mathrm{x}(G,H)$ when $G=G_{n,p}^{(r)}$, the Erdős-Rényi random $r$-uniform hypergraph, and $H=C_{2\mathrm{\ell }}^{(r)}$, the $r$-uniform linear cycle of length $2\mathrm{\ell }$. The case of graphs ($r=2$) is a longstanding open problem that has been investigated by many researchers. We determine the order of magnitude of $\mathrm{e}\mathrm{x}(G_{n,p}^{(r)},C_{2\mathrm{\ell }}^{(r)})$ for all $r\ge 4$ and all $\mathrm{\ell }\ge 2$ up to polylogarithmic factors for all values of $p=p(n)$. Our proof is based on the container method and uses a balanced supersaturation result for linear even cycles which improves upon previous such results by Ferber-Mckinley-Samotij and Balogh-Narayanan-Skokan.