The threshold for stacked triangulations [article]

Eyal Lubetzky, Yuval Peled
2022 arXiv   pre-print
A stacked triangulation of a d-simplex 𝐨={1,...,d+1} (d≥ 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∼𝒴_d(n,p) contain the faces of a stacked triangulation of the d-simplex 𝐨, with its internal vertices labeled in [n]. In the language of bootstrap
more » ... n 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≥ 2, showing it is asymptotically (α_d n)^-1/d, where α_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.
arXiv:2112.12780v2 fatcat:ybnc74komfhppbahz42sftrrnm