Percolation analysis

This entry focuses on the application of percolation analysis to problems in statistical mechanics. For a general discussion see Refs.

Sites, bonds, and clusters

This topic concerns the analysis of connectivity of elements (sites) distributed in different positions of a given large system. Using some connectivity rules it is possible to define bonds between pairs of sites. These bonds can be used to build up clusters of sites. Two sites in a cluster can be connected directly by a bond between them or indirectly by one or more sequences of bonds between pairs of sites. The sites of the system can belong to different types (species in the chemistry language). Bonds are usually permitted only between near sites.

Connectivity rules

The connectivity rules (bonding criteria) that permit the creation of bonds can be of different nature: distance between sites, energetic interaction, types of sites, etc. In addition the bonding criteria can be deterministic or probabilistic. In the statistical mechanics applications one can find different bonding criteria, for example:

• Geometric distance: Two sites, i, j, are bonded if the distance between then satisfies: $r_{ij}<R_{p}$.

• Energetic criteria: Two sites i, j, has a bonding probability given by $b(r_{ij})=\max \left\{0,\exp \left[u_{ij}(r_{ij})\right]\right\}$

Percolation threshold

The sizes of the clusters of a given system depend on different factors: density and distribution of sites, bonding criteria (which could include the effect of temperature and energy interactions), etc.