User:Rziff/Rigidity percolation

Rigidity percolation is the phenomenon where disordered system become rigid to various forces or stresses when a sufficient number of connections between particles become formed. In analogy to the conventional percolation transition where long-range connectivity through a random system takes place when a sufficient number of connections between neighbors are made, the word percolation is used to describe this transition. A lattice-based model for rigidity percolation was introduced by M. F. Thorpe in 1983 and has been studied by many researchers ever since. The behavior of the model is very rich and has numerous applications to colloidal systems, gels, the glass transition, and polymerization.

Rigidity percolation thresholds

Lattice	z	Site percolation threshold	Bond percolation threshold
generic triangular	3	0.69755(30),	0.6602(3),

Critical exponents

Exponent	2d	3d	4d	5d	6d
$\alpha$	0.48(5),
$\beta$	0.175(20),
$\nu$	1.21(6),

These satisfy the following scaling relations.

Bethe and hierarchical lattices

On the Bethe lattice,

See also

• Percolation
• Percolation theory
• Percolation critical exponents