Entanglement in percolation

Abstract

We study finite and infinite entangled graphs in the bond percolation process in three dimensions with density p. After a discussion of the relevant definitions, the entanglement critical probabilities are defined. The size of the maximal entangled graph at the origin is studied for small p, and it is shown that this graph has radius whose tail decays at least as fast as exp(-αn/log n); indeed, the logarithm may be replaced by any iterate of logarithm for an appropriate positive constant α. We explore the question of almost sure uniqueness of the infinite maximal open entangled graph when p is large, and we establish two relevant theorems. We make several conjectures concerning the properties of entangled graphs in percolation. © London Mathematical Society 2000.