The longest increasing subsequence of Brownian separable permutons

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 Brownian–type permutations, i.e., permutations \(\sigma_n\) sampled from the Brownian separable permutons \(\mu_p\) for \(p\in(0,1)\). We show that

where \(\alpha=\alpha(p)\in(1/2,1)\)  is the unique solution to an equation involving a number-theoretic function and \(X=X(p)\) is a non-deterministic and a.s.\ positive and finite random variable, which is a measurable function of the Brownian separable permuton \(\mu_p\). 

 

Based on this joint work with Arka Adhikari, Thomas Budzinski, William Da Silva and Delphin Sénizergues.