The Percolation Threshold for a Honeycomb Lattice

Percolation  Way of studying lattice disorder  Ignores physical and chemical properties  Way of defining geometric constants  Way of studying lattice disorder  Ignores physical and chemical properties  Way of defining geometric constants

Types of Percolation  Site Percolation - if two adjacent sites exist, then the bond between them exists.  Bond Percolation - If a bond exists, both sites must exist.  Site Percolation - if two adjacent sites exist, then the bond between them exists.  Bond Percolation - If a bond exists, both sites must exist.

Example The Honeycomb Lattice

Types of Percolation  Type of percolation will effect the probability associated with generating a specific cluster size  Our focus will be on site percolation.  Type of percolation will effect the probability associated with generating a specific cluster size  Our focus will be on site percolation.

Infinite Clusters  Infinite cluster is defined as a cluster that spans from one end to the other of an infinite lattice.  Not all probabilities will give this.  Define the percolation threshold p c  Infinite cluster is defined as a cluster that spans from one end to the other of an infinite lattice.  Not all probabilities will give this.  Define the percolation threshold p c

Percolation Threshold  The percolation threshold is defined as the probability of a site existing, where we see an infinite cluster for the first time.  Few exact results. Most turn to numeric methods.  The percolation threshold is defined as the probability of a site existing, where we see an infinite cluster for the first time.  Few exact results. Most turn to numeric methods.

Concerns with Monte Carlo  Mean value? Min Value?  Several conflicting results  Seems we can only do as good as an upper and lower bound.  Mean value? Min Value?  Several conflicting results  Seems we can only do as good as an upper and lower bound.

A Method to Calculate Pc If we simply had a line of a sites stretching to infinity, the probability of getting an infinite cluster would be given by… P = p c L p c must be 1 to see an infinite cluster If we simply had a line of a sites stretching to infinity, the probability of getting an infinite cluster would be given by… P = p c L p c must be 1 to see an infinite cluster

A Method to Calculate Pc If we consider lattices that have more connectivity, we would need to consider the number of ways a particular L could be realized. P = N 1 p c1 L1 + N 2 p c2 L2 + … If we consider lattices that have more connectivity, we would need to consider the number of ways a particular L could be realized. P = N 1 p c1 L1 + N 2 p c2 L2 + …

A Method to Calculate Pc To start, lets consider only the shortest length as the would give the smallest value of p c. By careful enumerating and counting we can come up with N and L as a function of m. To start, lets consider only the shortest length as the would give the smallest value of p c. By careful enumerating and counting we can come up with N and L as a function of m.

A Method to Calculate Pc Then L=2m-1 and N=m2 m. So we get P = m2 m p c (2m-1) = m(2p c 2 ) m / p c If 2p c 2 < 1 then P=0 Then L=2m-1 and N=m2 m. So we get P = m2 m p c (2m-1) = m(2p c 2 ) m / p c If 2p c 2 < 1 then P=0

A Method to Calculate Pc Thus, when this is equal to 1, we first get an infinite cluster which is the definition of p c. Hence, Pc = 2 (-1/2) =.7071 Which is consistent with the Monte Carlo Thus, when this is equal to 1, we first get an infinite cluster which is the definition of p c. Hence, Pc = 2 (-1/2) =.7071 Which is consistent with the Monte Carlo

Triangle lattice If you go through the same method, but use the triangular lattice, you get the exact result of.5 which is the excepted result!

Issues With other lattice  Square lattice doesn’t work because the shortest distance is a straight line.  Inverted honeycomb doesn’t work because it is also a straight line.  Generalization will need to include more probabilities.  Square lattice doesn’t work because the shortest distance is a straight line.  Inverted honeycomb doesn’t work because it is also a straight line.  Generalization will need to include more probabilities.

For the Future  Try to generalize into a broad equation.  Use to solve harder lattices.  Can we relate bond percolation? Perhaps a correlation between the two?  Try to generalize into a broad equation.  Use to solve harder lattices.  Can we relate bond percolation? Perhaps a correlation between the two?

Thanks Dedicated to the memory of Carlos Busser Aug Dec 2007 Dedicated to the memory of Carlos Busser Aug Dec 2007