CMSA Combinatorics, Physics, and Probability: The threshold for stacked triangulations
SEMINARS, CMSA EVENTS
Yuval Peled - Hebrew University of Jerusalem
This notion of percolation can be viewed as a simplification of simple-connectivity. Namely, a stacked triangulation of a triangle is obtained by repeatedly subdividing an inner face into three faces.
We ask: for which $p$ does the random simplicial complex Y_2(n,p) contain, for every triple $xyz$, the faces of a stacked triangulation of $xyz$ whose internal vertices are arbitrarily labeled in [n].
We consider this problem in every dimension d>=2, and our main result identifies a sharp probability threshold for percolation, showing it is asymptotically (c_d*n)^(-1/d), where c_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.
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