Decomposing random permutations into order-isomorphic subpermutations

Two permutations \sigma and \pi are \ell-similar if they can be decomposed into subpermutations \sigma(1), ... , \sigma(\ell) and \pi(1), ... , \pi(\ell) such that \sigma(i) is order-isomorphic to \pi(i) for all i \in [\ell ]. Recently, Dudek, Grytczuk, and Rucinski Variations on twins in permutations, Electron. J. Combin., 28 (2021), P3.19. posed the problem of determining the minimum \ell for which two permutations chosen independently and uniformly at random are \ell-similar. We show that two such permutations are O(n1/3 log11/6(n))-similar with high probability, which is tight up to a polylogarithmic factor. Our result also generalizes to simultaneous decompositions of multiple permutations.