Power-law bounds for increasing subsequences in Brownian separable permutons and homogeneous sets in Brownian cographons

Abstract

The Brownian separable permutons are a one-parameter family – indexed by p∈(0,1) – of universal limits of random constrained permutations. We show that for each p∈(0,1), there are explicit constants 1/2<α⁎(p)≤β⁎(p)<1 such that the length of the longest increasing subsequence in a random permutation of size n sampled from the Brownian separable permuton is between nα⁎(p)−o(1) and nβ⁎(p)+o(1) with probability tending to 1 as n→∞. In the symmetric case p=1/2, we have α⁎(p)≈0.812 and β⁎(p)≈0.975. We present numerical simulations which suggest that the lower bound α⁎(p) is close to optimal in the whole range p∈(0,1).Our results work equally well for the closely related Brownian cographons. In this setting, we show that for each p∈(0,1), the size of the largest clique (resp. independent set) in a random graph on n vertices sampled from the Brownian cographon is between nα⁎(p)−o(1) and nβ⁎(p)+o(1) (resp. nα⁎(1−p)−o(1) and nβ⁎(1−p)+o(1)) with probability tending to 1 as n→∞.Our proofs are based on the analysis of a fragmentation process embedded in a Brownian excursion introduced by Bertoin (2002). We expect that our techniques can be extended to prove similar bounds for uniform separable permutations and uniform cographs.