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percolation [2021/04/13 21:14]
Scott Lawrence [Two dimensions] 		percolation [2021/05/05 19:06] (current)
Scott Lawrence [Bethe lattice]
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 ===== Lattice theories ===== ===== Lattice theories =====
  
-Percolation can be defined as either bond or site percolation, and on any geometry of lattice. In both cases, by "lattice" we mean a(n infinite) undirected graph.+Percolation can be defined as either bond or site percolation, and on any geometry of lattice. In both cases, by "lattice" we mean an (infinite) undirected graph.
  
 Because percolation is not a unitary CFT, there is no Hamiltonian lattice theory that flows to percolation in the IR. All lattices in this section are spacetime lattices. Because percolation is not a unitary CFT, there is no Hamiltonian lattice theory that flows to percolation in the IR. All lattices in this section are spacetime lattices.
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 In six or more dimensions critical percolation is described by [[mean-field theory|mean-field theory]]. In six or more dimensions critical percolation is described by [[mean-field theory|mean-field theory]].
 +
 +===== Bethe lattice =====
 +
 +The Bethe lattice can be thought of as the $d\rightarrow\infty$ lattice. It is the infinite tree in which each vertex has the same number ($z$ --- called the coordination number) of neighbors. Note that $z=1$ is invalid, and $z=2$ is just a one-dimensional lattice.
 +
 +As with many statistical models, percolation can be exactly solved on a Bethe lattice: see for instance [[https://web.mit.edu/ceder/publications/Percolation.pdf|these notes]]. The critical exponent is $\beta = 1$; the percolation threshold is $p_c = (z-1)^{-1}$.
 ===== Related theories ===== ===== Related theories =====
  
   * [[Directed percolation]]   * [[Directed percolation]]
  
 +
 +===== External links =====
 +  * [[https://en.wikipedia.org/wiki/Percolation_theory|Wikipedia]]
Last modified: 2021/04/13 21:14
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