Definition of the Subject
Percolation models were introduced in the 1950s by Broadbent and Hammersley [1] to model theflow of fluid in a random medium. Since both terms, fluid and medium, may be broadly interpreted, percolation has a wide variety ofapplications, including thermal phase transitions, oil flow in sandstone, and the spread of epidemics. An important motivation for the development ofpercolation models was to provide an alternative to diffusion models, in which the randomness was associated with the fluid while the medium is relativelyhomogeneous. Since percolation models associate the randomness with the medium, it is possible for the fluid either to become trapped or to flowinfinitely far. This presence of a phase transition is an important reason for the importance of percolation models. The percolation threshold isa critical probability in the percolation model which corresponds to the phase transition point. Since the emphasis in percolation...
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Abbreviations
- Archimedean lattices:
-
A regular tiling is a tiling of the plane which consists entirely of regular polygons. (A regular polygon is one in which all side lengths are equal and all interior angles are equal.) An Archimedean lattice is the graph of vertices and edges of a regular tiling which is vertex‐transitive, i. e., for every pair of vertices, u and v, there is a graph isomorphism that maps u to v. There are exactly 11 Archimedean lattices. A notation for Archimedean lattices, which can also serve as a prescription for constructing them, is given in Grünbaum and Shephard [5]. Around any vertex (since all are equivalent, by vertex‐transitivity), starting with the smallest polygon touching the vertex, list the number of edges of the successive polygons around the vertex. For convenience, an exponent is used to indicate that a number of successive polygons have the same size.
- Bond percolation:
-
In a bond percolation model, a random subgraph is formed from an infinite graph G by retaining each edge of G with probability p, independently of all other edges.
- Dual graph:
-
A graph is planar if it may be drawn in the plane with no edges intersecting except at their endpoints, thus dividing the plane into faces. Every planar graph G has a dual graph, denoted here by \( { D(G) } \). \( { D(G) } \) may be constructed by placing a vertex of \( { D(G) } \) in each face of G and connecting two vertices of \( { D(G) } \) by an edge if the corresponding faces in G share a common edge. Note that \( { D(D(G)) = G } \).
- Line graph:
-
The line graph, \( { L(G) } \), of a graph G is constructed by placing a vertex of \( { L(G) } \) on each edge of G and connecting two vertices of \( { L(G) } \) if the corresponding edges of G share a common endpoint.
- Matching graphs:
-
A pair of matching graphs may be constructed from an underlying planar graph. Select a set F of faces of the graph. Construct a graph G by adding an edge in each face of F between any pair of vertices that are not already connected by an edge. Construct the matching graph \( { M(G) } \) of G by adding an edge between any pair of vertices in each face not in F that are not already connected by an edge. Note that \( { M(M(G)) = G } \).
- Percolation threshold:
-
In a percolation model with parameter p, there is a retention probability \( { p_\mathrm{c} } \), called the percolation threshold, above which the random subgraph contains an infinite connected component and below which all connected components are finite.
- Periodic graph:
-
A periodic graph is an infinite graph that can be represented in d‑dimensional space so that it is invariant under translations by all integer linear combinations of a fixed basis.
- Site percolation:
-
In a site percolation model, a random graph is formed from an infinite graph G by retaining each vertex of G with probability p, independently of all other vertices. An edge of G is retained in the random graph if both its endpoint vertices are retained.
Bibliography
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Books and Reviews
Bollobás B, Riordan O (2006) Percolation. Cambridge University Press, Cambridge
Grimmett G (1999) Percolation, 2nd edn. Springer, Berlin
Hughes BD (1996) Random Walks and Random Environments, vol 2: Random Environments. Oxford Science Publications, Oxford
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Wierman, J.C. (2009). Percolation Thresholds, Exact. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_390
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