The Percolation Threshold for a Honeycomb Lattice

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The Percolation Threshold for a Honeycomb Lattice
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Percolation

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

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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.

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ExampleThe Honeycomb Lattice
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Types of Percolation

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

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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 pc

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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.

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Concerns with Monte Carlo

• Mean value? Min Value?
• Several conflicting results
• Seems we can only do as good as an upper and
  lower bound.

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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 pcL
• pc must be 1 to see an infinite cluster

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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 N1pc1L1 N2pc2L2

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A Method to Calculate Pc

• To start, lets consider only the shortest length
  as the would give the smallest value of pc.
• By careful enumerating and counting we can come
  up with N and L as a function of m.

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A Method to Calculate Pc

• Then L2m-1 and Nm2m.
• So we get
• P m2mpc(2m-1) m(2pc2)m / pc
• If 2pc2 lt 1 then P0

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A Method to Calculate Pc

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

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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!

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Issues With other lattice

• Square lattice doesnt work because the shortest
  distance is a straight line.
• Inverted honeycomb doesnt work because it is
  also a straight line.
• Generalization will need to include more
  probabilities.

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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?

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Thanks

• Dedicated to the memory of
• Carlos Busser
• Aug 2007 - Dec 2007