On Bond Percolation in the Infinite Knight Graph
College
College of Arts and Sciences
Department
Mathematics
Faculty Mentor
Dr. Thomas Polaski
Abstract
For a graph G=(V,E), let Gp=(V,Bin(E,p)) where Bin(E,p) keeps edges from E with probability p independently (and discards an edge with probability 1-p). For a locally finite graph G (i.e. |V| is infinite and the degree of each vertex is finite), let C(p) be the event that Gp contains an infinite path. Our objective is to identify pc, the critical probability, which has the property that if p > pc, then Pr(C(p)) = 1 and if p < pc, then Pr(C(p)) = 0 (that pc exists is a standard fact in the study of bond percolation). Now suppose we take G to be the infinite knight graph which has vertex set in the integer grid such that (x0, y0)~(x1,y1) if and only if |x1-x0|+|y1-y0|=3 and x0 is not equal to x1 and y0 is not equal to y1. We allow B to be the subgraph created by placing the knight at (0,0) and only allowing the knight to move horizontally two units and one unit vertically from each position. We are interested in the case when H is taken to be the union of B and the graph created by reflecting B over the line y=x. In this case, we obtain a nontrivial upper and lower bound on pc for H, the former via coupling with bound percolation on the integer grid and the latter by an appropriate union bound.
Honors Thesis Committee
Thomas Polaski, Ph.D.; Kristen Abernathy, Ph.D.; Trent Kull, Ph.D.
Previously Presented/Performed?
11th Annual Regional Mathematics and Statistics Conference, University of North Carolina, Greensboro, November 2015
Southern Regional Honors Council Conference, Orlando, Florida, March 2016
Start Date
22-4-2016 1:25 PM
End Date
22-4-2016 1:40 PM
On Bond Percolation in the Infinite Knight Graph
DiGorgio Campus Center, Room 220
For a graph G=(V,E), let Gp=(V,Bin(E,p)) where Bin(E,p) keeps edges from E with probability p independently (and discards an edge with probability 1-p). For a locally finite graph G (i.e. |V| is infinite and the degree of each vertex is finite), let C(p) be the event that Gp contains an infinite path. Our objective is to identify pc, the critical probability, which has the property that if p > pc, then Pr(C(p)) = 1 and if p < pc, then Pr(C(p)) = 0 (that pc exists is a standard fact in the study of bond percolation). Now suppose we take G to be the infinite knight graph which has vertex set in the integer grid such that (x0, y0)~(x1,y1) if and only if |x1-x0|+|y1-y0|=3 and x0 is not equal to x1 and y0 is not equal to y1. We allow B to be the subgraph created by placing the knight at (0,0) and only allowing the knight to move horizontally two units and one unit vertically from each position. We are interested in the case when H is taken to be the union of B and the graph created by reflecting B over the line y=x. In this case, we obtain a nontrivial upper and lower bound on pc for H, the former via coupling with bound percolation on the integer grid and the latter by an appropriate union bound.
