The threshold for stacked triangulations

TL;DR

This paper determines the threshold probability for the appearance of stacked triangulations in random simplicial complexes, linking geometric combinatorics with bootstrap percolation in hypergraphs.

Contribution

It identifies the exact asymptotic threshold for stacked triangulations in the Linial--Meshulam model for all dimensions d ≥ 2.

Findings

Threshold is asymptotically (_d n)^{-1/d} for each dimension d.

Uses second moment method in supercritical regime.

Employs algebraic shifting in subcritical regime.

Abstract

A \emph{stacked triangulation} of a $d$-simplex $\mathbf{o}=\{1,\ldots,d+1\}$ ($d\geq 2$) is a triangulation obtained by repeatedly subdividing a $d$-simplex into $d+1$ new ones via a new vertex (the case $d=2$ is known as an Appolonian network). We study the occurrence of such a triangulation in the Linial--Meshulam model, i.e., for which $p$ does the random simplicial complex $Y\sim \mathcal{Y}_d(n,p)$ contain the faces of a stacked triangulation of the $d$-simplex $\mathbf{o}$, with its internal vertices labeled in $[n]$. In the language of bootstrap percolation in hypergraphs, it pertains to the threshold for $K_{d+2}^{d+1}$, the $(d+1)$-uniform clique on $d+2$ vertices. Our main result identifies this threshold for every $d\geq 2$, showing it is asymptotically $(\alpha_d n)^{-1/d}$, where $\alpha_d$ is the growth rate of the Fuss--Catalan numbers of order $d$. The proof hinges on a second moment argument in the supercritical regime, and on Kalai's algebraic shifting in the subcritical regime.