A 1-independent bond percolation model on a graph is a probability distribution on the spanning subgraphs of in which, for all vertex-disjoint sets of edges and , the states (i.e. present or not present) of the edges in are independent of the states of the edges in . Such models typically arise in renormalisation arguments applied to independent percolation models, or percolation models with finite range dependencies. A 1-independent model is said to percolate if the random subgraph has an infinite component with positive probability. In 2012 Balister and Bollob\’as defined to be the supremum of those for which there exists a 1-independent bond percolation model on in which each edge is present in the random subgraph with probability at least but which does not percolate. A fundamental and challenging problem in this area is to determine, or give good bounds on, the value of when is the lattice graph . Since , it is also of interest to establish the value of .
In this talk we will present a significantly improved upper bound for this limit as well as improved upper and lower bounds for . We will also show that with high confidence we have for large and discuss some open problems concerning 1-independent models on other graphs.
This is joint work with Tom Johnston and Alex Scott.