We establish a novel bijective encoding that represents permutations as forests of decorated (or enriched) trees. This allows us to prove local convergence of uniform random permutations from substitution-closed classes satisfying a criticality constraint. It also enables us to reprove and strengthen permuton limits for these classes in a new way, that uses a semi-local version of Aldous’ skeleton decomposition for size-constrained Galton–Watson trees.
"A decorated tree approach to random permutations in substitution-closed classes." Electron. J. Probab. 25 1 - 52, 2020. https://doi.org/10.1214/20-EJP469