# Universal limits of substitution-closed permutation classes

## TL;DR

This paper investigates the limiting behavior of uniform random permutations within substitution-closed classes, revealing a universal limit under certain conditions and identifying two other regimes with distinct limiting objects.

## Contribution

It introduces a framework linking permutation class generating series to permuton limits, including a universal limit and regimes related to stable trees, using singularity analysis.

## Key findings

- The limit depends on the generating series of simple permutations.
- Under mild conditions, the limit is a deformation of the Brownian separable permuton.
- Two other regimes with different limiting objects are identified.

## Abstract

We consider uniform random permutations in proper substitution-closed classes and study their limiting behavior in the sense of permutons. The limit depends on the generating series of the simple permutations in the class. Under a mild sufficient condition, the limit is an elementary one-parameter deformation of the limit of uniform separable permutations, previously identified as the Brownian separable permuton. This limiting object is therefore in some sense universal. We identify two other regimes with different limiting objects. The first one is degenerate; the second one is nontrivial and related to stable trees. These results are obtained thanks to a characterization of the convergence of random permutons through the convergence of their expected pattern densities. The limit of expected pattern densities is then computed by using the substitution tree encoding of permutations and performing singularity analysis on the tree series.