In the context of percolation theory, 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 not to depend on microscopic details like the lattice structure or whether site or bond percolation is considered. This article deals with the critical exponents of random percolation.
Contents
- Description
- Self similarity at the percolation threshold
- Critical behavior close to the percolation threshold
- Hyperscaling relations
- Relations based on displaystyle sigma tau
- Relations based on d f displaystyle dtextfnu
- Exponents for standard percolation on a non trivial planar lattice Weighted planar stochastic lattice WPSL
- References
Percolating systems have a parameter
Description
In the behavior of thermodynamic or configurational systems near a critical point or a continuous phase transition, the system become fractal and the behavior of many quantities are 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.
Critical exponents exist for a variety of observables, but most of them are linked to each other by exponent (or scaling) relations. Only a few of them are independent, and it is a matter of taste what the fundamental exponents are. One choice is the set
Self-similarity at the percolation threshold
Percolation clusters become self-similar precisely at the threshold density
The fractal dimension
The Fisher exponent
The probability for two sites separated by a distance
The exponent
The minimum or chemical distance exponent
Critical behavior close to the percolation threshold
The approach to the percolation threshold is governed by power laws again, which hold asymptotically close to
The exponent
Off criticality, only finite clusters exist up to a largest cluster size
The density of clusters (number of clusters per site)
The strength or weight of the percolating cluster
The divergence of the mean cluster size
Hyperscaling relations
Relations based on { σ , τ } {\displaystyle \{\sigma ,\tau \}}
Relations based on { d f , ν } {\displaystyle \{d_{\text{f}},\nu \}}
Exponents for standard percolation on a non-trivial planar lattice (Weighted planar stochastic lattice (WPSL))
Note that it has been claimed that the numerical values of exponents of percolation depends only on the dimension of lattice. However, percolation on WPSL is an exception in the sense that albeit it is two dimensional yet it does not belong to the same universality where all the planar lattices belong.