Anti-Ramsey threshold of cycles

For graphs $G$ and $H$, let $G \overset{\mathrm{rb}}{{\longrightarrow}} H$ denote the property that for every proper edge colouring of $G$ there is a rainbow copy of $H$ in $G$. Extending a result of Nenadov, Person, \v{S}kori\'{c} and Steger [J. Combin. Theory Ser. B 124 (2017),1-38], we determine the threshold for $G(n,p) \overset{\mathrm{rb}}{{\longrightarrow}} C_\ell$ for cycles $C_\ell$ of any given length $\ell \geq 4$.