In a recent work, Allen, Böttcher, Hàn, Kohayakawa, and Person provided a first general analogue of the blow-up lemma applicable to sparse (pseudo)random graphs thus generalising the classic tool of Komlós, Sárközy, and Szemerédi. Roughly speaking, they showed that with high probability in the random graph \(G_{n, p}\) for \(p \geqslant C(\log n/n)^{1/\Delta }\) , sparse regular pairs behave similarly as complete bipartite graphs with respect to embedding a spanning graph H with \(\Delta (H) \leqslant \Delta\) . However, this is typically only optimal when \(\Delta \in \{2,3\}\) and H either contains a triangle ( \(\Delta = 2\) ) or many copies of \(K_4\) ( \(\Delta = 3\) ). We go beyond this barrier for the first time and present a sparse blow-up lemma for cycles \(C_{2k-1}, C_{2k}\) , for all \(k \geqslant 2\) , and densities \(p \geqslant Cn^{-(k-1)/k}\) , which is in a way best possible. As an application of our blow-up lemma we fully resolve a question of Nenadov and Škorić regarding resilience of cycle factors in sparse random graphs.
