Scaling limits of permutation classes with a finite specification: A dichotomy

We consider uniform random permutations in classes having a finite combinatorial specification for the substitution decomposition. These include (but are not limited to) all permutation with number of simple permutations. Our goal is to study their limiting behavior sense permutons. The limit depends on structure restricted families largest growth rate. When it strongly connected, two cases occur. If associated system equations linear, permuton deterministic X-shape. Otherwise, Brownian separable permuton, object that already appeared as most substitution-closed classes, among which Moreover these results can be combined some non connected cases. To prove our result, we use characterization convergence permutons by subpermutations. Key steps study, via trees, marked elements inducing given pattern, and singularity analysis corresponding generating functions.