3:00 p.m., Monday (Jan. 21)
Math Annex 1100
Alexander E. Holroyd
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.