On the Brownian separable permuton

TL;DR

This paper characterizes the structure of the Brownian separable permuton, revealing its fractal, self-similar nature, and connecting it to Brownian excursions and continuum random trees, with explicit density calculations.

Contribution

It provides a detailed description of the permuton's structure, including its support, Hausdorff dimension, and self-similarity, along with explicit density formulas and connections to Brownian trees.

Findings

Permuton is the pushforward of Lebesgue measure on a Brownian excursion graph.

Support is totally disconnected with Hausdorff dimension one.

Explicit density function of the averaged permuton is derived.

Abstract

The Brownian separable permuton is a random probability measure on the unit square, which was introduced by Bassino, Bouvel, F\'eray, Gerin, Pierrot (2016) as the scaling limit of the diagram of the uniform separable permutation as size grows to infinity. We show that, almost surely, the permuton is the pushforward of the Lebesgue measure on the graph of a random measure-preserving function associated to a Brownian excursion whose strict local minima are decorated with i.i.d. signs. As a consequence, its support is almost surely totally disconnected, has Hausdorff dimension one, and enjoys self-similarity properties inherited from those of the Brownian excursion. The density function of the averaged permuton is computed and a connection with the shuffling of the Brownian continuum random tree is explored.