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

TL;DR

This work establishes sharp power-law bounds for the length of the longest increasing subsequence in random permutations drawn from Brownian separable permutons and the size of the largest homogeneous sets in Brownian cographons. The authors develop a fragmentation-based framework embedded in a Brownian excursion and hyper-precise probabilistic estimates via a Markovian selection rule to derive explicit exponents α_*(p) and β^*(p) that bound LIS between n^{α_*(p)-o(1)} and n^{β^*(p)+o(1)}. In the symmetric case p=1/2, the bounds yield α_*(p) ≈ 0.812 and β^*(p) ≈ 0.975, with simulations suggesting α_*(p) is near-optimal across p∈(0,1); analogous results hold for the largest homogeneous sets in Brownian cographons, translating LIS exponents to clique and independent set sizes. The methods connect Brownian-type continuum limits to discrete permutation and graph models, offering a pathway to transfer bounds to uniform separable permutations and uniform cographs via coupling arguments and highlighting deep links with fragmentation theory and Erdős–Hajnal-type phenomena in a probabilistic continuum setting.

Abstract

The Brownian separable permutons are a one-parameter family -- indexed by $p\in(0,1)$ -- of universal limits of random constrained permutations. We show that for each $p\in (0,1)$, there are explicit constants $1/2 < α_*(p) \leq β^*(p) < 1$ such that the length of the longest increasing subsequence in a random permutation of size $n$ sampled from the Brownian separable permuton is between $n^{α_*(p) - o(1)}$ and $n^{β^*(p) + o(1)}$ with probability tending to 1 as $n\to\infty$. In the symmetric case $p=1/2$, we have $α_*(p) \approx 0.812$ and $β^*(p)\approx 0.975$. We present numerical simulations which suggest that the lower bound $α_*(p)$ is close to optimal in the whole range $p\in(0,1)$. Our results work equally well for the closely related Brownian cographons. In this setting, we show that for each $p\in (0,1)$, the size of the largest clique (resp. independent set) in a random graph on $n$ vertices sampled from the Brownian cographon is between $n^{α_*(p) - o(1)}$ and $n^{β^*(p) + o(1)}$ (resp. $n^{α_*(1-p) - o(1)}$ and $n^{β^*(1-p) + o(1)}$) with probability tending to 1 as $n\to\infty$. 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.

Figures (11)

• Figure 1: Top: The diagram of three large permutations (in black) sampled from the Brownian separable permuton with parameter $p=0.2,0.5,0.9$ (from left to right). In red we highlighted one longest increasing subsequence. Bottom: The adjacency matrix of three large graphs (ones are plotted in black) sampled from the Brownian cographon with parameter $p=0.2,0.5,0.9$ (from left to right). In red we highlighted one largest homogeneous set. In the first two samples it is an independent set, while in the third case it is a clique.
• Figure 2: Left: The plot of our bounds from \ref{['thm:upper_lower_perm']} as functions of $p\in(0,1)$: $\alpha_*(p)$ is in blue and $\beta^*(p)$ is in red. Right: Some numerical values of the bounds $\alpha_*(p)$ and $\beta^*(p)$.
• Figure 3: The diagram shows the range of values $(\rho,q)\in [-1,1]\times(0,1)$ for the parameters of the skew Brownian permutons. At the bottom, in green, we have the Mallows permutons. At the top, in red, we have the Brownian separable permutons. In blue, close to the center, we have the Baxter permuton. Various models of random permutations which are known to converge in the permuton sense to the skew Brownian permutons are indicated between rounded parentheses. Finally, various models of planar maps which are connected to the skew Brownian permutons are indicated between squared parentheses.
• Figure 4: A sketch for the notation introduced in \ref{['sec: strategy lower bound']}.
• Figure 5: A sketch for the notation introduced for the self-similar interval fragmentation $\mathfrak{F}_0=(\mathfrak{F}_0(h),h\ge 0)$. In red we highlighted the intervals in $\mathfrak{F}_0(h)$ and the corresponding sub-excursions of $\mathfrak{e}$ above level $h$. The interval $I^{t}(h)$ in $\mathfrak{F}_0(h)$ containing $t$ has length $F^{t}(h)$ in $\mathfrak{F}$.

Theorems & Definitions (49)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Corollary 1.4
• Conjecture 1.5
• Conjecture 1.6
• Conjecture 1.7
• Remark 1.8
• Remark 1.9
• Remark 1.10