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On the Brownian separable permuton

Part of: Combinatorial probability Combinatorics

Published online by Cambridge University Press:  24 October 2019

Mickaël Maazoun
Mickaël Maazoun*
Affiliation:
Université de Lyon – École Normale Supérieure de Lyon – UMPA UMR 5669 CNRS, 46 allée d’Italie, 69364 Lyon, France
*
Email: [email protected]
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Abstract

The Brownian separable permuton is a random probability measure on the unit square, which was introduced by Bassino, Bouvel, Féray, Gerin and 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 independent and identically distributed 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.

MSC classification

Primary: 60C05: Combinatorial probability
Secondary: 05A05: Permutations, words, matrices
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 29 , Issue 2 , March 2020 , pp. 241 - 266
Copyright
© Cambridge University Press 2019

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