VCU Discrete Mathematics Seminar Bootstrap Percolation Thresholds in Plane Tilings using Regular Polygons Prof Dan Cranston VCU! Wednesday, Feb. 21 1:00-1:50 4145 Harris Hall We study bootstrap percolation, which is an example of a cellular automa- ton, sometimes called a 0-player game. We fix a positive integer k and start with a plane graph T , in which some faces are “infected”. Once a face is infected, it remains so forever. If a face, f, is uninfected, but has at least k infected neighbors, then f becomes infected. The percolation percolation thresh- old is the largest integer k such that if we infect each face independently with probability 1/2, then with probability at least 1/2 eventually the whole graph becomes infected. We consider bootstrap percolation in tilings of the plane by regular polygons. A vertex type in such a tiling is the (cyclic) order of the faces that meet a common vertex. Let T denote the set of plane tilings T by regular polygons such that if T contains one instance of a vertex type, then it contains infinitely many instances of that type. We show that no tiling in T has threshold 4 or more. Further, we show that the only tilings in T with threshold 3 are four of the Archimedean lattices. Finally, we describe a large subclass of T with threshold 2. This is joint work with Neal Bushaw. For the DM seminar schedule, see: http://www.people.vcu.edu/ ~ dcranston/DM-seminar
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VCU Discrete Mathematics Seminar
Bootstrap Percolation Thresholdsin Plane Tilings using Regular Polygons
Prof Dan CranstonVCU!
Wednesday, Feb. 211:00-1:50
4145 Harris Hall
We study bootstrap percolation, which is an example of a cellular automa-ton, sometimes called a 0-player game. We fix a positive integer k and startwith a plane graph T , in which some faces are “infected”. Once a face isinfected, it remains so forever. If a face, f, is uninfected, but has at least k
infected neighbors, then f becomes infected. The percolation percolation thresh-old is the largest integer k such that if we infect each face independently withprobability 1/2, then with probability at least 1/2 eventually the whole graphbecomes infected.
We consider bootstrap percolation in tilings of the plane by regular polygons.A vertex type in such a tiling is the (cyclic) order of the faces that meet acommon vertex. Let T denote the set of plane tilings T by regular polygonssuch that if T contains one instance of a vertex type, then it contains infinitelymany instances of that type. We show that no tiling in T has threshold 4 ormore. Further, we show that the only tilings in T with threshold 3 are fourof the Archimedean lattices. Finally, we describe a large subclass of T withthreshold 2.
This is joint work with Neal Bushaw.
For the DM seminar schedule, see:
http://www.people.vcu.edu/~dcranston/DM-seminar
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