Title:

Percolation beyond connectivity

Percolation models are of interest both for the wide range of their physical applications and for the mathematical challenges which they present. The basic model is as follows. Starting from an infinite connected graph such as the hypercubic lattice, each edge is declared 'open' with probability p or 'closed' otherwise, independently of all others. The standard theory is primarily concerned with the existence (or not) of infinite connected components of the graph of open edges, [2]. Various extensions of the basic model have been studied in detail, [1, 2, 5]. In this work we extend the model in a direction which has received less attention: rather than studying connected components, we consider other graph properties analogous to connectivity. We explore this idea with particular reference to two such properties which have important physical applications, [3, 4]: entanglement and rigidity. Roughly speaking, the meaning of these terms is as follows. A graph in threedimensional space is entangled if it cannot be 'pulled apart' when the edges are regarded as physical connections made of elastic. A graph is rigid if it cannot be 'deformed' when the edges are regarded as solid rods which can pivot at the vertices. We formalise these intuitive notions for both finite and infinite graphs. In the case of infinite graphs this involves overcoming interesting challenges which are related to the issue of boundary conditions. Having defined entanglement and rigidity formally, we consider entangled and rigid graphs in the percolation model. We prove that (under suitable conditions) there is a genuine phase transition for each, occurring at critical probabilities which differ from the usual critical probability for connectivity percolation. For p below the appropriate critical probability, we explore the size of finite entangled or rigid components. For p greater than the appropriate critical probability we study the question of uniqueness of the infinite entangled or rigid component. We prove several relevant theorems including uniqueness for entanglement for large p, and uniqueness for rigidity for almost all p.
