# 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 G_{p}=(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 G_{p} contains an infinite path. Our objective is to identify p_{c}, the critical probability, which has the property that if p > p_{c}, then Pr(C(p)) = 1 and if p < p_{c}, then Pr(C(p)) = 0 (that p_{c} 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 (x_{0}, y_{0})~(x_{1},y_{1}) if and only if |x_{1}-x_{0}|+|y_{1}-y_{0}|=3 and x_{0} is not equal to x_{1} and y_{0} is not equal to y_{1}. 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 p_{c} 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 G_{p}=(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 G_{p} contains an infinite path. Our objective is to identify p_{c}, the critical probability, which has the property that if p > p_{c}, then Pr(C(p)) = 1 and if p < p_{c}, then Pr(C(p)) = 0 (that p_{c} 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 (x_{0}, y_{0})~(x_{1},y_{1}) if and only if |x_{1}-x_{0}|+|y_{1}-y_{0}|=3 and x_{0} is not equal to x_{1} and y_{0} is not equal to y_{1}. 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 p_{c} for H, the former via coupling with bound percolation on the integer grid and the latter by an appropriate union bound.