TY - JOUR
T1 - The threshold for stacked triangulations
AU - Lubetzky, Eyal
AU - Peled, Yuval
N1 - Publisher Copyright:
© The Author(s) 2022.
PY - 2023/10/1
Y1 - 2023/10/1
N2 - 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, that is, for which p does the random simplicial complex Y ∼ Yd(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 percolation in hypergraphs, it pertains to the threshold for Kd+1d+2, the (d + 1)-uniform clique on d + 2 vertices. Our main result identifies this threshold for every d ≥ 2, showing it is asymptotically (αdn)−1/d, where αd 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.
AB - 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, that is, for which p does the random simplicial complex Y ∼ Yd(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 percolation in hypergraphs, it pertains to the threshold for Kd+1d+2, the (d + 1)-uniform clique on d + 2 vertices. Our main result identifies this threshold for every d ≥ 2, showing it is asymptotically (αdn)−1/d, where αd 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.
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U2 - 10.1093/imrn/rnac276
DO - 10.1093/imrn/rnac276
M3 - Article
AN - SCOPUS:85142189211
SN - 1073-7928
VL - 2023
SP - 16296
EP - 16335
JO - International Mathematics Research Notices
JF - International Mathematics Research Notices
IS - 19
ER -