PhD Defense by Bounghun Bock

Event Details
  • Date/Time:
    • Thursday May 2, 2019
      1:30 pm - 3:30 pm
  • Location: Skiles 006
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Summaries

Summary Sentence: Percolation Theory: The complement of the infinite cluster & The acceptance profile of the invasion percolation

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Percolation Theory: The complement of the infinite cluster & The acceptance profile of the invasion percolation

Series: 

Dissertation Defense

Thursday, May 2, 2019 - 1:30pm

1.5 hours (actually 80 minutes)

Location: 

Skiles 006

Speaker: 

Bounghun Bock

,  

Georgia Tech

,  

bbock3@gatech.edu

Organizer: 

Bounghun Bock

In independent bond percolation  with parameter p, if one removes the vertices of the infinite cluster (and incident edges), for which values of p does the remaining graph contain an infinite cluster? Grimmett-Holroyd-Kozma used the triangle condition to show that for d > 18, the set of such p contains values strictly larger than the percolation threshold pc. With the work of Fitzner-van der Hofstad, this has been reduced to d > 10. We reprove this result by showing that for d > 10 and some p>pc, there are infinite paths consisting of "shielded"' vertices --- vertices all whose adjacent edges are closed --- which must be in the complement of the infinite cluster. Using numerical values of pc, this bound can be reduced to d > 7. Our methods are elementary and do not require the triangle condition.

Invasion percolation is a stochastic growth model that follows a greedy algorithm. After assigning i.i.d. uniform random variables (weights) to all edges of d-dimensional space, the growth starts at the origin. At each step, we adjoin to the current cluster the edge of minimal weight from its boundary. In '85, Chayes-Chayes-Newman studied the "acceptance profile"' of the invasion: for a given p in [0,1], it is the ratio of the expected number of invaded edges until time n with weight in [p,p+dp] to the expected number of observed edges (those in the cluster or its boundary) with weight in the same interval. They showed that in all dimensions, the acceptance profile an(p) converges to one for p<pc and to zero for p>pc. In this paper, we consider an(p) at the critical point p=pc in two dimensions and show that it is bounded away from zero and one as n goes to infinity.

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Graduate Studies

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Phd Defense
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  • Created By: Tatianna Richardson
  • Workflow Status: Published
  • Created On: Apr 12, 2019 - 10:35am
  • Last Updated: Apr 12, 2019 - 10:35am