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DTSTART;TZID=America/New_York:20220503T093000
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CREATED:20240214T072025Z
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UID:10002557-1651570200-1651573800@cmsa.fas.harvard.edu
SUMMARY:The threshold for stacked triangulations
DESCRIPTION:Abstract: Consider a bootstrap percolation process that starts with a set of `infected’ triangles $Y \subseteq \binom{[n]}3$\, and a new triangle f gets infected if there is a copy of K_4^3 (= the boundary of a tetrahedron) in which f is the only not-yet infected triangle.\nSuppose that every triangle is initially infected independently with probability p=p(n)\, what is the threshold probability for percolation — the event that all triangles get infected? How many new triangles do get infected in the subcritical regime? \nThis 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.\nWe 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]. \nWe 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. \nThe proof hinges on a second moment argument in the supercritical regime\, and on Kalai’s algebraic shifting in the subcritical regime. \nJoint work with Eyal Lubetzky.
URL:https://cmsa.fas.harvard.edu/event/5-3-2022-cmsa-combinatorics-physics-and-probability-seminar/
LOCATION:Hybrid
CATEGORIES:Combinatorics Physics and Probability
ATTACH;FMTTYPE=image/png:https://cmsa.fas.harvard.edu/media/CMSA-Combinatorics-Physics-and-Probability-Seminar-05.03.22-1.png
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