On the Brownian separable permuton

Mickaël Maazoun (1)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées

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

Mots clés: Brownian tree; Permutation patterns; Brownian excursion; Brownian CRT; Separable permutations; Permuton

Domaines: Probabilités [math.PR]; Combinatoire [math.CO]

Mickaël Maazoun. On the Brownian separable permuton. Combinatorics, Probability and Computing, 2020, 29 (2), pp.241-266. ⟨10.1017/S0963548319000300⟩. ⟨ensl-01651215v1⟩