Percolation surface critical behavior concerns the influence of surfaces on the critical behavior of percolation.
Background
Percolation is the study of connectivity in random systems, such as electrical conductivity in random conductor/insulator systems, fluid flow in porous media, gelation in polymer systems, etc. At a critical fraction of connectivity or porosity, long-range connectivity can take place, leading to long-range flow. The point where that connectivity takes place is called the percolation threshold, and considerable amount of work has been undertaken in finding those critical values for systems of various geometries, and the mathematical behavior of observables near that point. This leads to the study of critical behavior and the percolation critical exponents. These exponents allow one to describe the behavior as the threshold is approached. The behavior of the percolating network near a surface will be different from that in the main part of a system, called the "bulk." For example, exact at the percolation threshold, the percolating network in the system is a fractal with large voids and a ramified structure. The surface interrupts this structure, so the percolating cluster is less likely to come in contact to the surface. As an example, consider a lattice system of bond percolation (percolation along the bonds or edges of the lattice). If the lattice is cubic in nature, and p {\displaystyle p} is the probability that a bond is occupied (conducting), then the percolation threshold is known to be p c = 0.311608... {\displaystyle p_{c}=0.311608...} . At the surface, the lattice becomes a simple square lattice, where the bond threshold p c {\displaystyle p_{c}} is simply 1/2. Therefore, when the bulk of the system is at its threshold, the surface is way below its threshold, and the only way to have long-range connections along the surface is to have a path that goes from the surface to the bulk, conduction through the fractal percolation network, and then a path back to the surface again. This occurs with a different critical behavior as in the bulk, and is different from the critical behavior of a two-dimensional surface at its threshold. In the most common model for surface critical behavior in percolation, all bonds are assigned with the same probability p {\displaystyle p} , and the behavior is studied at the bulk p c {\displaystyle p_{c}} , with a value of 0.311608 in this case. In an other model for surface behavior, the surface bonds are made occupied with a different probability p s {\displaystyle p_{s}} , while the bulk is kept at the normal bulk value. When p ( s ) {\displaystyle p^{(s)}} is increased to a higher value, a new "special" critical point is reached p c ( s ) {\displaystyle p_{c}^{(s)}} , which has a different set of critical exponents.
Surface phase transitions
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