Topological connectivity of random permutation complexes

TL;DR

This paper investigates the topological connectivity properties of simplicial complexes derived from random permutations, establishing bounds on the probability that these complexes are not topologically r-connected.

Contribution

It introduces a novel probabilistic model linking permutation groups to topological properties of associated complexes and provides bounds on connectivity failure probabilities.

Findings

Probability bounds depend on n and r, with specific logarithmic factors.

As n grows, the likelihood of not being r-connected diminishes at a rate involving (log n)^r/n.

The results quantify how permutation-induced complexes become topologically connected as size increases.

Abstract

Let $\mathbb{S}_n$ denote the symmetric group on $[n]=\{1,\ldots,n\}$ with the uniform probability measure. For a permutation $\pi \in \mathbb{S}_n$ let $X_{\pi}$ denote the simplicial complex on the vertex set $[n]$ whose simplices are all $\{i_0,\ldots, i_m\} \subset [n]$ such that $i_0<\cdots<i_m$ and $\pi(i_0)<\cdots < \pi(i_m)$. For $r \geq 0$ let $p_r(n)$ denote the probability that $X_{\pi}$ is not topologically $r$-connected for $\pi \in \mathbb{S}_n$. It is shown that for fixed $r \geq 0$ there exist constants $0<C_r, C_r' < \infty$ such that \[ C_r \frac{(\log n)^r}{n} \leq p_r(n) \leq C_r' \frac{(\log n)^{2r}}{n}. \]