Critical exponents of characteristic functions

Universal percolating claster

You see below site percolation to the left and continuum percolation in random potential landscape to the right near percolation thresholds.
Trying several realizations you can make sure that both percolating (infinite) clusters are very similar and do not depend on small scale stucture of the lattices.

Critical exponents

In the critical region |p - pc| << 1 the percolation functions are observed to obey power-law dependances
    P(p) ~ (p-pc)b,     S(p) ~ |p-pc|-g,     L(p) ~ |p-pc|-n,
where b, g, n are critical exponents.

The percolation threshold is a quantity which varies greatly from lattice to lattice. However, the critical exponents do not depend on the details of lattice geometry. They are the same for all lattices of the same dimensionality D, i.e. the critical exponents are dimensional invariants. For D = 2 the critical exponents are b = 0.14, g = 2.4, n = 1.35. Some important relations between these critical exponents follow from scaling laws, e.g.
    Dn = 2b + g.

It is amazing that the critical exponents do not depend on the nearest neighbor number. You see below "square" lattices with 4 neighbors (or 8 with diagonal sites), 6 and 3 neighbors.

    4                      6
  |   |           | / | /        \ / \ /
- o - o -       - o - o -       - o - o -
  |   |         / | / | /  ~   \ / \ /
- o - o -       - o - o -     - o - o -
  |   |         / | / |        / \ / \

                      3
      |     /     |          \     /     \ 
  .   o - o   .   o -         o - o       o -
    /     |     /            /     \     /
- o   .   o - o   .   ~   - o       o - o 
  |     /     |              \     /     \
  o - o   .   o - o           o - o       o -

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updated 21 Apr 2004