In the context of the physical and mathematical theory of percolation, a percolation transition is characterized by a set of universal critical exponents, which describe the fractal properties of the percolating medium at large scales and sufficiently close to the transition.
The exponents are universal in the sense that they only depend on the type of percolation model and on the space dimension.
They are expected to not depend on microscopic details such as the lattice structure, or whether site or bond percolation is considered.
This article deals with the critical exponents of random percolation.
, the mean cluster size goes to infinity and the percolation transition takes place.
, various quantities either diverge or go to a constant value by a power law in
While the exponent of that power law is generally the same on both sides of the threshold, the coefficient or "amplitude" is generally different, leading to a universal amplitude ratio.
Thermodynamic or configurational systems near a critical point or a continuous phase transition become fractal, and the behavior of many quantities in such circumstances is described by universal critical exponents.
Percolation theory is a particularly simple and fundamental model in statistical mechanics which has a critical point, and a great deal of work has been done in finding its critical exponents, both theoretically (limited to two dimensions) and numerically.
Only a few of them are independent, and the choice of the fundamental exponents depends on the focus of the study at hand.
So-called correction exponents extend these sets, they refer to higher orders of the asymptotic expansion around the critical point.
Percolation clusters become self-similar precisely at the threshold density
for sufficiently large length scales, entailing the following asymptotic power laws: The fractal dimension
relates how the mass of the incipient infinite cluster depends on the radius or another length measure,
, normalized by the total volume (number of lattice sites).
is connected with the leading correction to scaling, which appears, e.g., in the asymptotic expansion of the cluster-size distribution,
The elastic backbone [2] has the same fractal dimension as the shortest path.
, which describes the scaling of the mass M of a critical cluster within a chemical distance
The chemical distance can also be thought of as a time in an epidemic growth process, and one also defines
Also related to the minimum dimension is the simultaneous growth of two nearby clusters.
[6] The fractal dimension of the random walk on an infinite incipient percolation cluster is given by
The approach to the percolation threshold is governed by power laws again, which hold asymptotically close to
, and the cluster-size distribution is smoothly cut off by a rapidly decaying function,
is continuous at the threshold but its third derivative goes to infinity as determined by the exponent
The probability a point at a surface belongs to the percolating or infinite cluster for
[8] The mean size of finite clusters connected to a site in the surface is
This system is referred to as "1 + 1 dimensional DP" where the two dimensions are thought of as space and time.
are the transverse (perpendicular) and longitudinal (parallel) correlation length exponents, respectively.
is the exponent corresponding to the behavior of the survival probability as a function of time:
) is the exponent corresponding to the behavior of the average number of visited sites at time