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BEGIN:VEVENT
UID:8ec7a7227dc6773d3e4384f7ce273721
CATEGORIES:Discrete Math
CREATED:20220413T092105
SUMMARY:Threshold for stacked triangulations
LOCATION:Hill Center Room 705
DESCRIPTION:<p>Abstract: The K_4^3 bootstrap percolation process is defined as follows:
  start with an initial set of "infected'' triangles Y, where each of the {n
  choose 3} triangles with vertices [n]={1,2,…,n} appears independently with
  probability p; then repeatedly add to it a new triangle {a,b,c} if there e
 xists a tetrahedron in which this is the only missing face (i.e. if for som
 e x the 3 triangles {a,b,x},{a,x,c},{x,b,c} are already infected). Let Y_in
 fty denoted the final state of the process. What is the critical probabilit
 y p(n) so that Y_infty would typically contain a specific triangle {1,2,3}?
  How many triangles would Y_infty typically have below that threshold? When
  would Y_infty typically contain all triangles? <br />Equivalently, a stack
 ed triangulation of a triangle with labels in [n], a.k.a. an Appolonian Net
 work, is one obtained by repeatedly subdividing a triangle {a,b,c} into 3 n
 ew triangles {a,b,x},{a,x,c},{x,b,c} via a label x in [n]. The above questi
 ons would amount to asking, e.g., about the critical probability so that th
 e random simplicial complex Y_2(n,p) would typically contain the faces of a
  stacked triangulation of every triangle {a,b,c}.<br />We consider these qu
 estions for a general dimension d geq 2, and our results identify the criti
 cal threshold p_c for stacked triangulations: we show that p_c is asymptoti
 cally (C(d) n)^(-1/d), where the constant C(d) is the growth rate of the Fu
 ss--Catalan numbers of order d. The proof hinges on a second moment argumen
 t in the supercritical regime, and on Kalai's algebraic shifting in the sub
 critical regime.<br />Joint work with Yuval Peled.</p>
CONTACT:Eyal Lubetzky (Courant Inst. at NYU)
DTSTAMP:20260827T103212
DTSTART;TZID=America/New_York:20220418T140000
DTEND;TZID=America/New_York:20220418T150000
SEQUENCE:0
TRANSP:OPAQUE
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