Entanglement and Rigidity in Percolation

Percolation theory plays a central role in the study of disordered systems. In the basic model, each edge of the d-dimensional cubic lattice is independently declared "open" with probability p, or "closed" with probability 1-p. The standard theory is concerned with the connected components of the resulting graph of open edges.

Recently, there has been progress on certain natural extensions of the percolation model, in which connectivity is replaced with some other graph property. Two properties of interest are entanglement and rigidity. Loosely speaking, a graph is said to be entangled if it cannot be "pulled apart" when the edges are regarded as connections in three-dimensional space; a graph is said to be rigid if it cannot be "deformed" when the edges are regarded as solid bars which can pivot at the vertices.

The study of entanglement and rigidity in percolation leads to exciting mathematical challenges. Certain standard percolation results (for example, relating to existence of phase transitions, and uniqueness of infinite clusters) require entirely new methods of proof. In other cases there are new types of behaviour not found in the connectivity case. Examples of the latter occur in relation to boundary conditions.