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percolation [2021/04/04 05:37]
Scott Lawrence 		percolation [2021/05/05 19:06] (current)
Scott Lawrence [Bethe lattice]
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 ====== Percolation ====== ====== Percolation ======
  
-Percolation is a [[Logarithmic CFT|logarithmic CFT]]. Among other things, this means it is not unitary.+{{tag>2d 3d 4d 5d logarithmic non-unitary}}
  
-This page focuses on "standard percolation" --- see [[#Related theories]] below for alternative definitions that yield different CFTs.+Percolation is a [[tag:logarithmic|logarithmic CFT]]. Among other things, this means it is not unitary. 
 + 
 +This page focuses on "standard percolation" --- see [[#related theories]] below for alternative definitions that yield different CFTs. 
 + 
 +===== 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 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. 
 + 
 +Usually percolation is defined on a square lattice. Many other possibilities exist, with the same critical behavior. 
 + 
 +==== Site percolation ==== 
 + 
 +==== Bond percolation ==== 
 + 
 + 
 +===== Computational methods ===== 
 + 
 +==== Lattice ==== 
 + 
 +The most straightforward computational method is to instantiate a random lattice of some finite size and compute cluster sizes. This gives a single sample, and the procedure can be repeated as much as desired to improve the statistics. Note that although this is a Monte Carlo method, there is no (non-trivial) Markov chain. In other words, there is no "thermalization" or "mixing time" needed in the simulation. 
 + 
 +==== Epsilon expansion ====
  
 ===== One dimension ===== ===== One dimension =====
  
-Percolation in one dimension is, unsurprisingly, exactly solvable.+Percolation in one dimension is, unsurprisingly, exactly solvable. Clearly no critical cluster can exist unless $p=1$, so that's the critical point. Working out the average cluster size at arbitrary $p$ can be an exercise.
  
 ===== Two dimensions ===== ===== Two dimensions =====
  
 +For bond percolation on a 2-dimensional square lattice, $p_c = 0.5$ is known exactly.
 ===== Six or more dimensions ===== ===== Six or more dimensions =====
  
 +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]]
 +
 +
 +===== External links =====
 +  * [[https://en.wikipedia.org/wiki/Percolation_theory|Wikipedia]]
Last modified: 2021/04/04 05:37
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