# Probability and Mathematical Physics Seminar

#### Long increasing subsequences in Brownian-type permutations

**Speaker:**
Jacopo Borga, Stanford University

**Location:**
Warren Weaver Hall 1302

**Date:**
Friday, November 10, 2023, 11:10 a.m.

**Synopsis:**

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 sampled from the Brownian separable permutons. We show that there are explicit constants 1/2<α<β<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^{α–o(1)} and n^{β+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.