The Brownian limit of separable permutations

TL;DR

This paper investigates the asymptotic behavior of pattern occurrences in random separable permutations, revealing their convergence to a Brownian-based limit called the Brownian separable permuton, thus connecting permutation patterns with stochastic processes.

Contribution

It introduces the concept of a Brownian separable permuton as the limit of uniform random separable permutations, linking permutation patterns to Brownian excursions.

Findings

Asymptotic distribution of pattern occurrences described by Brownian excursion

Convergence of random separable permutations to a Brownian separable permuton

Establishment of a probabilistic limit for pattern-avoiding permutations

Abstract

We study random uniform permutations in an important class of pattern-avoiding permutations: the separable permutations. We describe the asymptotics of the number of occurrences of any fixed given pattern in such a random permutation in terms of the Brownian excursion. In the recent terminology of permutons, our work can be interpreted as the convergence of uniform random separable permutations towards a "Brownian separable permuton".