On the Brownian separable permuton

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 i...

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Published inCombinatorics, probability & computing Vol. 29; no. 2; pp. 241 - 266
Main Author Maazoun, Mickaël
Format Journal Article
LanguageEnglish
Published Cambridge Cambridge University Press 01.03.2020
Cambridge University Press (CUP)
Subjects
Combinatorics
Mathematics
Permutations
Probability
Self-similarity
Signs
Brownian tree
Permutation patterns
Brownian CRT
Separable permutations
Permuton
Brownian excursion
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ISSN0963-5483
1469-2163
DOI10.1017/S0963548319000300

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Summary: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.
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ISSN:0963-5483
1469-2163
DOI:10.1017/S0963548319000300