Long increasing subsequences in Brownian-type permutations

What is the behavior of the longest increasing subsequence of a uniformly random permutation? 
Its length is of order \(2n^{1/2}\) plus Tracy–Widom fluctuations of order \(n^{1/6}\). Its scaling limit is the directed geodesic of the directed landscape
This talk discusses how this behavior changes dramatically when one looks at universal Browniantype permutations, i.e., permutations sampled from the Brownian separable permutons. We show that there are explicit constants \(1/2 < \alpha< \beta < 1\) such that the length of the longest increasing subsequence in a random permutation of size \(n\) sampled from the Brownian separable permutons is between \(n^{\alpha – o(1)}\) and \(n^{\beta + o(1)}\) with high probability. We present numerical simulations which suggest that the lower bound is close to optimal. 
Our proofs are based on the analysis of a fragmentation process embedded in a Brownian excursion introduced by Bertoin (2002). 
If time permits, we conclude by discussing some conjectures for permutations sampled from the skew Brownian permutons, a model of universal permutons generalizing the Brownian separable permutons: here, the longest increasing subsequences should be closely related with some models of random directed metrics on planar maps. 
Based on joint work with William Da Silva and Ewain Gwynne.


Institute for Advanced Study, Princeton, New Jersey (USA), Probability seminar.


The University of British Columbia, British Columbia (Canada), Probability seminar.


The University of Washington, Washington state (USA), Probability seminar.

4 thoughts on “Long increasing subsequences in Brownian-type permutations”

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